Methods for improving performance and reliability of biosensors

ABSTRACT

The present invention relates to a predictive-kinetic method for use with data processing of a sensor-generated signal, as well as, microprocessors and monitoring systems employing such a predictive-kinetic method. Data from a transient region of a signal is used with suitable models and curve-fitting methods to predict the signal that would be measured for the system at the completion of the reaction. The values resulting from data processing of sensor response using the methods of the present invention are less sensitive to measurement variables.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is related to U.S. Provisional PatentApplications Serial Nos. 60/204,397, filed May 16, 2000, and 60/244,078,filed Oct. 27, 2000, from both of which priority is claimed under 35 USC§119(e)(1), and which applications are incorporated herein by referencein their entireties.

TECHNICAL FIELD

[0002] The present invention relates generally to the use of apredictive-kinetic method to reduce the effects of measurement variableson results obtained using analyte sensors. The invention includes amethod and device for measuring the concentration of target analytespresent in a biological system, for example, a mammalian subject. Moreparticularly, the invention relates to methods, microprocessors, andmonitoring systems for predicting an amount or concentration of ananalyte using a series of measurements obtained from a monitoring systemand employing a predictive-kinetic algorithm.

BACKGROUND OF THE INVENTION

[0003] Measurement and data-processing approaches related to enzymereaction-based biosensors have historically been based on evaluation ofnon-equilibrium steady-state responses. Two limitations of such analysesinclude the following: (i) loss of sensitivity as the substrateconcentration approaches and exceeds the corresponding Michaelisconstant of the enzyme immobilized on the sensor, and (ii) adverseinfluences on measured values due to changes in experimental variablesthat influence (a) rates of chemical reactions, and (b) physicalprocesses that control the steady state response.

[0004] Similar problems have been encountered in conventionalkinetic-based methods when they are applied to enzymatic determinationsof analytes in homogeneous solutions (Chen, W., et al., AnalyticaChimica Acta 388:231-241, 1999). Results of such analyses generally havelimited ranges of linearity and are influenced by experimental variablesthat affect enzyme activity. Steady-state data-analysis methods appliedto enzyme reaction-based sensors are influenced by variables that affectrates of reaction and rates of mass transport. However, application ofinitial-rate methods using enzymes in homogenous solution (i.e.,kinetic-based solution methods) tend to be influenced only by variablesthat affect rates of reactions provided the solutions are well-stirred.

[0005] A variety of measurement and data-processing approaches have beenused in attempts to reduce or eliminate problems in homogenous solutionmeasurement of analyte concentrations including, but not limited to, thefollowing. Engh, et al., (Anal. Chem. 60:545, 1988), used alternativeapplications of a rate-based approach and showed improvement in theruggedness of enzymatic methods but also demonstrated that the methodsdid little to improve the sensitivity at high concentrations ofsubstrate. For homogenous solution analyses, a two-rate method(Wentzell, P. D., et al, Anal. Chem. 58:2851, 1986) andpseudoequilibrium methods (Meiling, G. E., et al., Anal. Chem. 50:1611,1978; Harris, R. C., Clin. Chem. 29:2079, 1983) have demonstrated thepotential to reduce dependencies on experimental variables to a similardegree as has been seen with equilibrium methods. Further, the two-rateand pseudoequilibrium methods, when used in this way, appear to maintainhigh sensitivity for analyte concentrations above Michaelis constants.

[0006] Two-rate and pseudoequilibrium methods (based on homogenoussystem methods) have been applied to enzyme-based biosensor methods todetermine if these methods could be adapted to biosensors such thatmeasurement improvements would be seen which were similar to thoseachieved in homogenous solution (Chen, et al., Analytica Chimica Acta388:231-241, 1999; Wentzell, P. D., et al, Anal. Chem. 58:2851, 1986;Meiling, G. E., et al., Anal. Chem. 50:1611, 1978; Harris, R. C., Clin.Chem. 29:2079, 1983). The enzyme-based biosensor typically used in suchstudies consisted of an enzyme and an electron mediator immobilized onthe surface of a glassy-carbon electrode (e.g., Chen, et al., AnalyticaChimica Acta 388:231-241, 1999). Although some improvements inperformance characteristics of the enzyme-based biosensor were observed,both methods were shown to have limitations when applied to enzyme-basedbiosensor data.

SUMMARY OF THE INVENTION

[0007] The present invention relates to methods, microprocessors, andmonitoring systems for predicting a concentration or amount of ananalyte using measurements obtained from a monitoring system andemploying a predictive-kinetic algorithm. In one embodiment, the presentinvention relates to a method for measuring the amount or concentrationof an analyte present in a biological system. In the method, a samplecomprising the analyte of interest is transdermally extracted using asampling system, for example, by ionotophoresis, sonophoresis,laser-formed micro-holes and suction, where the sampling system is inoperative contact with a skin or mucosal surface of the biologicalsystem. Typically, frequent samples are obtained over time while thesampling system remains in operative contact with the surface. Ameasured signal is obtained, e.g., employing a sensing device, from theextracted analyte. Typically the measured signal is a response curvecomprising data points with respect to time. The measured signal isspecifically related to the amount or concentration of analyte, and theresponse curve comprises kinetic and equilibrium regions. At least onemathematical model comprising selected parameters is chosen where themodel describes the curve. In preferred embodiments, the mathematicalmodel is selected from the group consisting of a first order process,combined first order and zero order process, a parallel multiple firstorder process, a flux process, an n^(th) order process (where n does notequal one), and mixtures and combinations thereof.

[0008] The model and an error minimization method are iteratively usedto provide a predicted response curve corresponding to the measuredsignal response curve, wherein (i) the error minimization methodprovides a calculated error (e.g., chi-square) based on differencesbetween the predicted and measured signal response curves, and (ii) theestimating is iteratively performed until the calculated error betweenthe predicted and measured signal response curves falls within anacceptable range or until no further statistically significant change isseen in the calculated error. At this point, iterative estimation of theparameters is stopped. The iterative estimation and error minimizationresults in a predicted response curve corresponding to the measuredsignal response curve. The predicted response curve yields a predictedend-point value and a measurement correlated to the amount orconcentration of the analyte. For example, when the analyte sensor isdetecting current, the end-point value obtained using thepredictive-kinetic method typically represents a final background value.In one embodiment, the current is obtained from a system afterintroduction of the analyte (e.g., by transdermal extraction) andapplication of an appropriate potential. In a further example, afterintegration of a predicted response curve based on current, theend-point analyte-related value typically represents an area under thecurve. A background correction step may be performed prior tointegration, for example, the final background value obtained from thepredicted current response curve may be employed for backgroundsubtraction, the predicted current response curve integrated and theend-point analyte-related value determined.

[0009] Exemplary embodiments of the measured signal are current andcharge. The mathematical model may further comprise more than oneprocess and each process may comprise selected parameters. In a furtherembodiment, each process may be associated with a weighting factor. Inaddition, the mathematical model may comprise a zero-order process,and/or at least one quadratic or square root term. Backgroundsubtraction may also be performed on the measured signal, for example,before application of the predictive-kinetic methods.

[0010] In some embodiments of the present invention, the end-pointanalyte-related value is converted to an amount or concentration of theanalyte using, for example, a method comprising a calibration value(e.g., a ratio, a calibration point, a difference value, etc.).

[0011] Typically, at least two analyte samples are obtained and theircorresponding measured signal response curves analyzed to provide a“series of measurements.” In some embodiments of the present invention,conversion of the end-point or equilibrium analyte-related value tocorrespond to the amount or concentration of analyte can be carried onas each end-point analyte-related value is obtained, calculated togetherat the end, calculated in clusters, or any combination thereof.

[0012] In one aspect of the invention, at least three data points areobtained from the kinetic region of the curve, and these data points areused to estimate the half-life of the signal. The estimates of thehalf-life (t_(½)) may comprise, for example, estimating a rate constant(k) using a first order model. The obtaining of measured signal can thenbe carried out for a period of time determined based on the half-life,for example, the obtaining of measured signal can continue for a timeperiod corresponding to at least three half-lives of the signal.

[0013] In an alternative embodiment, obtaining the measured signal canbe carried out for a predetermined period of time. Such a defined timeperiod may be, for example, empirically determined.

[0014] The measured signal may be transformed in a variety of waysbefore estimation of the end-point analyte-related values using themathematical model, for example, the measured signal data can beintegrated. Integration can be performed, for example, with or withoutbackground correction of the original signal (e.g., using backgroundsubtraction, see below). In one embodiment, background subtraction isperformed by subtracting the predicted final background value from eachpoint making up the predicted response curve, the background correctedpredicted response curve is then integrated to obtain an end-pointanalyte-related value.

[0015] For different measurements in a series (i.e., for differentmeasured signal data curves obtained at different time points) differentmathematical models may be selected to estimate the end-point values.Alternately, all end-point values may be estimated using a singlemathematical model.

[0016] In one embodiment of the present invention, the mathematicalmodel comprises a first order process, for example, the first orderprocess may comprise the following:

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0017] where S_(o), S_(t), and S_(∞) are initial, intermediate, andend-point signals, k and t are the observed first-order rate constantand time, respectively.

[0018] In another embodiment of the present invention, the mathematicalmodel comprises a parallel multiple first order process, for example,the parallel multiple first order process may comprise the following:

S _(t) =S _(o)+(S ₂₈ ₁ −S _(o))*(1−e ^(−k1t))+(S _(∞2) −S _(o))*(1−e^(−k2t))+(S_(∞3) −S _(o))*(1 31 e ^(−k3t))+ . . .  (Eqn. 6A)

[0019] where S_(o) S_(t) are initial and intermediate signals, S₂₈ ₁,S_(∞2), S_(∞3), etc., are end-point signals (related to k₁, k₂, k₃,etc., respectively), k₁, k₂, k₃, etc., are the observed first-order rateconstants, and t is time. In this embodiment, the predicted end-pointvalue may be described by the following equation

S _(∞)=(_(S) _(∞1) +S _(∞2) +S _(∞3)+ . . . )+S_(o)  (Eqn. 6B).

[0020] Further, a change in the predicted end-point value relative tothe initial signal is described by the following equation:

ΔS _(∞)=(S _(∞1) +S _(∞2) +S _(∞3)+ . . . )  (Eqn. 6C).

[0021] The parallel multiple first order process may comprises thefollowing:

S _(t) =S _(o)+(S _(∞1) −S _(o))*(1−e ^(−k1t))+(S _(∞2) −S _(o))*(1−e^(−k2t))  (Eqn. 10)

[0022] where S_(o), and S_(t), are initial and intermediate signals,S_(∞1), and S_(∞2) are end-point signals (related to k₁ and k₂,respectively), k₁, k₂, and t are the observed first-order rate constantsand time. Further, a selected parallel multiple first order process mayfurther comprise at least one zero order process, for example, asfollows:

S _(t) =S _(o) +k _(o) t+(S _(∞1) −S _(o))*(1−e ^(k) ^(−k1t))+(S_(∞2) −S_(o))*(1−e ^(−k2t))+(S _(∞3) −S _(o))*(1−e ^(−k3t))  (Eqn. 16)

[0023] where S_(o), S_(t) are initial and intermediate signals, S_(∞1),S_(∞2), S_(∞3), are end-point signals (related to k₁, k₂, k₃,respectively), k₁, k₂, k₃, are the observed first-order rate constants,k_(o) is a zero order rate constant, and t is time.

[0024] Further, a selected parallel multiple first order process mayfurther comprise at least one quadratic or square root term.

[0025] In a further embodiment of a parallel multiple first orderprocess, for example, wherein the measured signal response curvecomprises a measurement of current over time, the parallel multiplefirst order process may comprise the following:

S _(t) =S ₀ +S ₁ *e ^(−k1*t) +S ₂ *e ^(−k2*t)+final_Bkgrd  Eqn. 20

[0026] where S₀ is response at t=0, t is time, S_(t) is a total signalat time t, S₁ and S₂ are signals at time t consistent with two processesassociated with apparent rate constants k₁ and k₂, and final_bkgrd is anestimated signal response at completion of a reaction used to obtain themeasured signal. In one embodiment, the area under the predictedresponse curve is obtained by integration. In a related embodiment,before the integration is performed the final_bkgrd value is used toperform a background subtraction correction of the predicted responsecurve and the measurement correlated to the amount or concentration ofglucose corresponds to the area under the predicted response curve.

[0027] In a further embodiment of a parallel multiple first orderprocess, for example, wherein the measured signal response curvecomprises a measurement of current over time, the parallel multiplefirst order process may comprise the following:

S _(t) =S ₁ *e ^(−k1*t) +S ₂*e^(−k2*t)+final_Bkgrd  Eqn. 21

[0028] where t is time, S_(t) is a total signal at time t, S₁ and S₂ aresignals at time t consistent with two processes associated with apparentrate constants k, and k₂, and final_bkgrd is an estimated signalresponse at completion of a reaction used to obtain the measured signal.As just described, the area under the predicted response curve may beobtained by integration. Further, before the integration is performedthe final_bkgrd value may be used to perform a background subtractioncorrection of the predicted response curve, and the measurementcorrelated to the amount or concentration of glucose corresponds to thearea under the predicted response curve.

[0029] In another aspect of the present invention, the mathematicalmodel comprises an n^(th) order process, for example, as follows:

S _(t) =S _(∞)(−/+)[k(n−1)*t(+/−)(S _(∞) −S_(o))^(1−n)]^(1/(1−n))  (Eqn. 8)

[0030] where S_(o), S_(t), and S_(∞) are initial, intermediate, andend-point signals, k and t are the observed rate constant and time, n isthe order of the process, where n does not equal 1, and for (−/+) thefirst function (−) is used for data that increase in magnitude as afunction of time, and the second function (+) is used for the reversecase, correspondingly for (+/−) the first function (+) is used for datathat increase in magnitude as a function of time, and the secondfunction (−) is used for the reverse case.

[0031] In yet a further aspect of the present invention, themathematical model comprises a flux model, for example, as follows:$\begin{matrix}{S_{t} = {S_{o} + {\left( {S_{\infty} - S_{o}} \right)\left\lbrack {1 + {2{\sum\limits_{i = 0}^{\infty}{\left( {- 1} \right)^{i}{\exp \left( {{- k_{i}}t} \right)}}}}} \right\rbrack}}} & \left( {{Eqn}.\quad 9} \right)\end{matrix}$

[0032] where S_(o), S_(t), and S_(∞), are initial, intermediate, andfinal (or end-point) signals, k_(i)=ki²π², k is the characteristicdiffusion rate constant, t is time, and i is a dummy-variable.

[0033] In one aspect of the present invention, for example, when atleast three data points are obtained from the kinetic region of themeasured signal response curve, these data points may be used toestimate the half-life of the measured signal. In one embodiment, theestimate of the half-life (t_(½)) further may comprise, estimating arate constant (k). In one embodiment, such an estimate is carried outusing a first order model, for example, as follows:

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0034] where S_(o), S_(t), and S_(∞) are initial, intermediate, andend-point signals, k and t are the observed first-order rate constantand time, respectively, wherein estimating the rate constant isperformed by a method comprising plotting the natural log of signal(S_(t)−S_(o)) over time, where the slope of the resulting linecorresponds to an estimate of k, and the half-life of the signal iscalculated by using t_(1/2)=ln 2/k.

[0035] In a further aspect of the present invention, the analyte, forexample, glucose, may be extracted by the sampling system into one ormore collection reservoirs to obtain a concentration of the glucose in areservoir. In one embodiment, the sampling system, comprising thecollection reservoirs, is in contact with the skin or mucosal surface ofthe subject and the analyte is extracted using an iontophoretic currentapplied to the skin or mucosal surface. The collection reservoir(s) maycomprise an enzyme composition, comprising an enzyme that reacts withthe extracted analyte, e.g., glucose, to produce an electrochemicallydetectable signal. In one aspect of the invention, wherein the analyteis glucose, the enzyme may be glucose oxidase.

[0036] In another embodiment, the present invention describes a methodfor compensating for an incomplete reaction involving the detection ofan analyte by predicting a background signal. This method also employsthe predictive-kinetic methods of the present invention, as describedherein.

[0037] The present invention also includes one or more microprocessorsprogrammed to perform the calculations of the predictive-kinetic methodsdescribed herein. Such microprocessors may be further programmed tocontrol associated devices, including, but not limited to, a samplingdevice, a sensing device, a power source, displays, etc.

[0038] The present invention also includes monitoring systems employingthe methods described herein for frequent measurement an analyte amountor concentration present in a biological system. In one aspect, amonitoring system of the present invention comprises a sampling devicefor extracting the analyte from the biological system into at least onecollection reservoir to obtain a concentration of the analyte in thereservoir. Typically, the collection reservoir is in contact with theskin or mucosal surface of the biological system. In one embodiment, theanalyte is extracted using an iontophoretic current applied to the skinor mucosal surface. The collection reservoir may comprise an enzyme, orenzymes, used to produce an electrochemically detectable signal(s)corresponding to the analyte(s) of interest. Signals are detected usinga sensing device. In a preferred embodiment the analyte comprisesglucose and the enzyme comprises glucose oxidase. One or moremicroprocessors are programmed to control, for example, the sampling,sensing, computations employing the predictive-kinetic methods describedherein, and displays of resulting values.

[0039] These and other embodiments of the present invention will readilyoccur to those of ordinary skill in the art in view of the disclosureherein.

BRIEF DESCRIPTION OF THE FIGURES

[0040]FIG. 1 is an exploded pictorial representation of components froman exemplary sampling system.

[0041]FIG. 2 illustrates kinetic and equilibrium regions of the responseof a biosensor to an analyte. The figure generically represents twoexemplary situations to which the predictive-kinetic methods of thepresent invention can be applied. In the figure, curve A corresponds,for example, to an amperometric (current) signal obtained from a sensor.The end-point value A, in this case, corresponds essentially to a finalbackground signal after depletion of the analyte-related signal. Thearea under curve A corresponds to a value related to analyte amount orconcentration. Curve B corresponds to an integrated form of curve A, oralternatively, a situation in which, for example, charge is directlymeasured by the sensor (instead of current). Accordingly, if curve A isa current signal response, then curve B is the corresponding chargesignal response curve. In this case the end-point value B corresponds toan end-point analyte-related value, that is, a value related to analyteamount or concentration (i.e., the area under curve A).

[0042]FIG. 3 shows experimental and fitted data for charge versus timeresponses using a 200 micromolar solution of glucose.

[0043]FIG. 4 shows experimental and fitted data for charge versus timeresponses using a 200 micromolar solution of glucose, where theexperimental data was monitored to completion.

[0044]FIG. 5 shows charge versus time responses for different glucoseconcentrations where the fitted lines were calculated using three halflives of the signal.

[0045]FIG. 6 presents a plot of predicted charge at three half lives ofthe signal versus the theoretical charge based on the glucoseconcentration used.

[0046]FIG. 7 presents a plot of a typical fit of a first order model tosignal from a sensor device.

[0047]FIG. 8 presents a plot of a typical fit of a parallel multiplefirst order model to signal from a sensor device.

[0048]FIG. 9 presents a plot of typical measurements for a non-diabeticsubject using data from a sensor device (employing predicted chargevalues based on a parallel multiple first order model fit to the dataand a fixed point method), as well as blood glucose measured by aconventional, invasive meter.

[0049]FIG. 10 presents a plot of the efficiency of error minimization(chi-square) using first order and parallel multiple first order modelsto fit the data. The parallel multiple first-order model better fits theresponse data and is not affected by analyte concentration. Thefirst-order model best fit the response data when analyte concentrationwas low. However, at high analyte concentration (e.g., between 3:00 and5:00 hours elapsed time in FIG. 10) the fit to the response data usingthe first-order model had higher associated error, as shown by the highchi-square values in this region.

[0050]FIG. 11 presents response curves for typical measurements from anon-diabetic subject using two data-processing methods, one using apredictive-kinetic method to determine glucose signal related charge(closed diamonds) and the second using a fixed point method to determineglucose signal related charge (closed squares). Data for blood glucoseamounts as determined using a OneTouch® (Johnson & Johnson, NewBrunswick, N.J.) device are presented in solid triangles with thereference axis being the right vertical axis.

[0051]FIG. 12 presents the ratio of predicted versus fixed-time pointsignal methods based on the data presented in FIG. 11.

[0052]FIG. 13 presents the formulae for some data processing models thatmay be useful in the practice of the present invention.

[0053]FIG. 14 presents typical time dependent responses of thebiographer glucose monitor to different concentrations of glucose (opensquares, 0.06 mM glucose; open triangles, 0.045 mM glucose; X, 0.03 mMglucose; *, 0.015 mM glucose; open diamond, 0.00888 mM glucose; and, +,0.00267 mM glucose).

[0054]FIG. 15 presents integrated responses from the data presented inFIG. 14, after background correction using the current at 405 seconds.

[0055]FIG. 16 presents fitted and experimental curves corresponding tobackground corrected experimental data from FIG. 15, wherein the curvefitting was carried out employing a predictive-kinetic method (Eqn. 19).

[0056]FIG. 17 presents fitted and experimental data for different levelsof glucose corresponding to data from FIG. 14, wherein the curve fitswere carried out employing a predictive-kinetic method (Eqn. 21). In thefigure, open squares, 0.06 mM glucose; open triangles, 0.045 mM glucose;X, 0.03 mM glucose; *, 0.015 mM glucose; open diamond, 0.00888 mMglucose; +, 0.00267 mM glucose; and the lines connecting the data pointscorrespond to the fitted curve predicted by the method.

[0057]FIG. 18 presents integrated responses from fitted current afterbackground correction using the predicted background current that wasobtained in FIG. 17, where the data fit range was 30-405 seconds.

[0058]FIG. 19 presents background current (nA) plotted as a function ofconcentration (mM). In the figure, open circles correspond to measuredbackground values at 405 seconds, closed circles correspond to predictedbackground values, the dark solid line corresponds to a linearregression of measured background values, and the light solid linecorresponds to a linear regression of predicted background values.

[0059]FIGS. 20A and 20B present schematic diagrams of, respectively,negative deviation (dashed line) from an ideal response (solid line),and positive deviation (dashed line) from an ideal response (solidline).

DETAILED DESCRIPTION OF THE INVENTION

[0060] All publications, patents and patent applications cited hereinare hereby incorporated by reference in their entireties.

[0061] 1. Definitions

[0062] It is to be understood that the terminology used herein is forthe purpose of describing particular embodiments only, and is notintended to be limiting. As used in this specification and the appendedclaims, the singular forms “a”, “an” and “the” include plural referentsunless the context clearly dictates otherwise. Thus, for example,reference to “a reservoir” includes a combination of two or more suchreservoirs, reference to “an analyte” includes mixtures of analytes, andthe like.

[0063] Unless defined otherwise, all technical and scientific terms usedherein have the same meaning as commonly understood by one of ordinaryskill in the art to which the invention pertains. Although other methodsand materials similar, or equivalent, to those described herein can beused in the practice of the present invention, the preferred materialsand methods are described herein.

[0064] In describing and claiming the present invention, the followingterminology will be used in accordance with the definitions set outbelow.

[0065] The term “microprocessor” refers to a computer processorcontained on an integrated circuit chip, such a processor may alsoinclude memory and associated circuits. A microprocessor may furthercomprise programmed instructions to execute or control selectedfunctions, computational methods, switching, etc. Microprocessors andassociated devices are commercially available from a number of sources,including, but not limited to, Cypress Semiconductor Corporation, SanJose, Calif.; IBM Corporation, White Plains, N.Y.; Applied MicrosystemsCorporation, Redmond, Wash.; Intel Corporation, Chandler, Ariz.; and,National Semiconductor, Santa Clara, Calif.

[0066] The terms “analyte” and “target analyte” are used to denote anyphysiological analyte of interest that is a specific substance orcomponent that is being detected and/or measured in a chemical,physical, enzymatic, or optical analysis—as long as thedetection/measurement is obtained over time (e.g., is time-dependent andprovides a response curve), the predictive-kinetic methods of thepresent invention can be applied. A detectable signal (e.g., a chemicalsignal or electrochemical signal) can be obtained, either directly orindirectly, from such an analyte or derivatives thereof. Furthermore,the terms “analyte” and “substance” are used interchangeably herein, andare intended to have the same meaning, and thus encompass any substanceof interest. In preferred embodiments, the analyte is a physiologicalanalyte of interest, for example, glucose, or a chemical that has aphysiological action, for example, a drug or pharmacological agent.

[0067] A “sampling device,” “sampling mechanism” or “sampling system”refers to any device and/or associated method for obtaining a samplefrom a biological system for the purpose of determining theconcentration of an analyte of interest. Such “biological systems”include any biological system from which the analyte of interest can beextracted, including, but not limited to, blood, interstitial fluid,perspiration and tears. Further, a “biological system” includes bothliving and artificially maintained systems. The term “sampling”mechanism refers to extraction of a substance from the biologicalsystem, generally across a membrane such as the stratum corneum ormucosal membranes, wherein said sampling is invasive, minimallyinvasive, semi-invasive or non-invasive. The membrane can be natural orartificial, and can be of plant or animal nature, such as natural orartificial skin, blood vessel tissue, intestinal tissue, and the like.Typically, the sampling mechanism is in operative contact with a“reservoir,” or “collection reservoir,” wherein the sampling mechanismis used for extracting the analyte from the biological system into thereservoir to obtain the analyte in the reservoir. Non-limiting examplesof sampling techniques include iontophoresis, sonophoresis (see, e.g.,International Publication No. WO 91/12772, published Sep. 5, 1991; U.S.Pat. No. 5,636,632), suction, electroporation, thermal poration, passivediffusion (see, e.g., International Publication Nos.: WO 97/38126(published Oct. 16, 1997); WO 97/42888, WO 97/42886, WO 97/42885, and WO97/42882 (all published Nov. 20, 1997); and WO 97/43962 (published Nov.27, 1997)), microfine (miniature) lances or cannulas, subcutaneousimplants or insertions, and laser devices (see, e.g., Jacques et al.(1978) J. Invest. Dermatology 88:88-93; International Publication WO99/44507, published Sep. 10, 1999 ; International Publication WO99/44638, published Sep. 10 1999; and International Publication WO99/40848, published Aug. 19, 1999). Iontophoretic sampling devices aredescribed, for example, in International Publication No. WO 97/24059,published Jul. 10, 1997; European Patent Application EP 0942 278,published Sep. 15, 1999; International Publication No. WO 96/00110,published Jan. 4, 1996; International Publication No. WO 97/10499,published Mar. 2, 1997; U.S. Pat. Nos. 5,279,543; 5,362,307; 5,730,714;5,771,890; 5,989,409; 5,735,273; 5,827,183; 5,954,685 and 6,023,629, allof which are herein incorporated by reference in their entireties.Further, a polymeric membrane may be used at, for example, the electrodesurface to block or inhibit access of interfering species to thereactive surface of the electrode.

[0068] The term “physiological fluid” refers to any desired fluid to besampled, and includes, but is not limited to, blood, cerebrospinalfluid, interstitial fluid, semen, sweat, saliva, urine and the like.

[0069] The term “artificial membrane” or “artificial surface,” refersto, for example, a polymeric membrane, or an aggregation of cells ofmonolayer thickness or greater which are grown or cultured in vivo or invitro, wherein said membrane or surface functions as a tissue of anorganism but is not actually derived, or excised, from a pre-existingsource or host.

[0070] A “monitoring system” refers to a system useful for obtainingfrequent measurements of a physiological analyte present in a biologicalsystem. Such a system typically includes, but is not limited to,sampling mechanism, sensing mechanism, and a microprocessor mechanism inoperative communication with the sampling mechanism and the sensingmechanism.

[0071] A “measurement cycle” typically comprises extraction of ananalyte from a subject, using, for example, a sampling device, andsensing of the extracted analyte, for example, using a sensing device,to provide a measured signal, for example, a measured signal responsecurve. A complete measurement cycle may comprise one or more sets ofextraction and sensing.

[0072] The term “frequent measurement” refers to a series of two or moremeasurements obtained from a particular biological system, whichmeasurements are obtained using a single device maintained in operativecontact with the biological system over a time period in which a seriesof measurements (e.g, second, minute or hour intervals) is obtained. Theterm thus includes continual and continuous measurements.

[0073] The term “subject” encompasses any warm-blooded animal,particularly including a member of the class Mammalia such as, withoutlimitation, humans and nonhuman primates such as chimpanzees and otherapes and monkey species; farm animals such as cattle, sheep, pigs, goatsand horses; domestic mammals such as dogs and cats; laboratory animalsincluding rodents such as mice, rats and guinea pigs, and the like. Theterm does not denote a particular age or sex and, thus, includes adultand newborn subjects, whether male or female.

[0074] The term “transdermal” includes both transdermal and transmucosaltechniques, i.e., extraction of a target analyte across skin, e.g.,stratum corneum, or mucosal tissue. Aspects of the invention which aredescribed herein in the context of “transdermal,” unless otherwisespecified, are meant to apply to both transdermal and transmucosaltechniques.

[0075] The term “transdermal extraction,” or “transdermally extracted”refers to any sampling method, which entails extracting and/ortransporting an analyte from beneath a tissue surface across skin ormucosal tissue. The term thus includes extraction of an analyte using,for example, iontophoresis (reverse iontophoresis), electroosmosis,sonophoresis, microdialysis, suction, and passive diffusion. Thesemethods can, of course, be coupled with application of skin penetrationenhancers or skin permeability enhancing technique such as varioussubstances or physical methods such as tape stripping or pricking withmicro-needles. The term “transdermally extracted” also encompassesextraction techniques which employ thermal poration, lasermicroporation, electroporation, microfine lances, microfine cannulas,subcutaneous implants or insertions, combinations thereof, and the like.

[0076] The term “iontophoresis” refers to a method for transportingsubstances across tissue by way of an application of electrical energyto the tissue. In conventional iontophoresis, a reservoir is provided atthe tissue surface to serve as a container of (or to provide containmentfor) material to be transported. Iontophoresis can be carried out usingstandard methods known to those of skill in the art, for example byestablishing an electrical potential using a direct current (DC) betweenfixed anode and cathode “iontophoretic electrodes,” alternating a directcurrent between anode and cathode iontophoretic electrodes, or using amore complex waveform such as applying a current with alternatingpolarity (AP) between iontophoretic electrodes (so that each electrodeis alternately an anode or a cathode). For example, see U.S. Pat. Nos.5,771,890 and 6,023,629 and PCT Publication No. WO 96/00109, publishedJan. 4, 1996.

[0077] The term “reverse iontophoresis” refers to the movement of asubstance from a biological fluid across a membrane by way of an appliedelectric potential or current. In reverse iontophoresis, a reservoir isprovided at the tissue surface to receive the extracted material, asused in the GlucoWatch® (Cygnus, Inc., Redwood City, Calif.) biographerglucose monitor (See, e.g., Tamada et al. (1999) JAMA 282:1839-1844).

[0078] “Electroosmosis” refers to the movement of a substance through amembrane by way of an electric field-induced convective flow. The termsiontophoresis, reverse iontophoresis, and electroosmosis, will be usedinterchangeably herein to refer to movement of any ionically charged oruncharged substance across a membrane (e.g., an epithelial membrane)upon application of an electric potential to the membrane through anionically conductive medium.

[0079] The term “sensing device,” “sensing mechanism,” or “biosensordevice” encompasses any device that can be used to measure theconcentration or amount of an analyte, or derivative thereof, ofinterest. Preferred sensing devices for detecting blood analytesgenerally include electrochemical devices, optical and chemical devicesand combinations thereof. Examples of electrochemical devices includethe Clark electrode system (see, e.g., Updike, et al., (1967) Nature214:986-988), and other amperometric, coulometric, or potentiometricelectrochemical devices, as well as, optical methods, for example UVdetection.

[0080] A “biosensor” or “biosensor device” includes, but is not limitedto, a “sensor element” that includes, but is not limited to, a“biosensor electrode” or “sensing electrode” or “working electrode”which refers to the electrode that is monitored to determine the amountof electrical signal at a point in time or over a given time period,which signal is then correlated with the concentration of a chemicalcompound. The sensing electrode comprises a reactive surface whichconverts the analyte, or a derivative thereof, to electrical signal. Thereactive surface can be comprised of any electrically conductivematerial such as, but not limited to, platinum-group metals (including,platinum, palladium, rhodium, ruthenium, osmium, and iridium), nickel,copper, and silver, as well as, oxides, and dioxides, thereof, andcombinations or alloys of the foregoing, which may include carbon aswell. Some catalytic materials, membranes, and fabrication technologiessuitable for the construction of amperometric biosensors are describedby Newman, J. D., et al.(1995) Analytical Chemistry 67:4594-4599.

[0081] The “sensor element” can include components in addition to thesensing electrode, for example, it can include a “reference electrode”and a “counter electrode.” The term “reference electrode” is used tomean an electrode that provides a reference potential, e.g., a potentialcan be established between a reference electrode and a workingelectrode. The term “counter electrode” is used to mean an electrode inan electrochemical circuit that acts as a current source or sink tocomplete the electrochemical circuit. Although it is not essential thata counter electrode be employed where a reference electrode is includedin the circuit and the electrode is capable of performing the functionof a counter electrode, it is preferred to have separate counter andreference electrodes because the reference potential provided by thereference electrode is most stable when it is at equilibrium. If thereference electrode is required to act further as a counter electrode,the current flowing through the reference electrode may disturb thisequilibrium. Consequently, separate electrodes functioning as counterand reference electrodes are preferred.

[0082] In one embodiment, the “counter electrode” of the “sensorelement” comprises a “bimodal electrode.” The term “bimodal electrode”typically refers to an electrode which is capable of functioningnon-simultaneously as, for example, both the counter electrode (of the“sensor element”) and the iontophoretic electrode (of the “samplingmechanism”) as described, for example, U.S. Pat. No. 5,954,685.

[0083] The terms “reactive surface,” and “reactive face” are usedinterchangeably herein to mean the surface of the sensing electrodethat: (1) is in contact with the surface of an ionically conductivematerial which contains an analyte or through which an analyte, or aderivative thereof, flows from a source thereof; (2) is comprised of acatalytic material (e.g., a platinum group metal, platinum, palladium,rhodium, ruthenium, or nickel and/or oxides, dioxides and combinationsor alloys thereof) or a material that provides sites for electrochemicalreaction; (3) converts a chemical signal (for example, hydrogenperoxide) into an electrical signal (e.g., an electrical current); and(4) defines the electrode surface area that, when composed of a reactivematerial, is sufficient to drive the electrochemical reaction at a ratesufficient to generate a detectable, reproducibly measurable, electricalsignal that is correlatable with the amount of analyte present in theelectrolyte.

[0084] An “ionically conductive material” refers to any material thatprovides ionic conductivity, and through which electrochemically activespecies can diffuse. The ionically conductive material can be, forexample, a solid, liquid, or semi-solid (e.g., in the form of a gel)material that contains an electrolyte, which can be composed primarilyof water and ions (e.g., sodium chloride), and generally comprises 50%or more water by weight. The material can be in the form of a hydrogel,a sponge or pad (e.g., soaked with an electrolytic solution), or anyother material that can contain an electrolyte and allow passage ofelectrochemically active species, especially the analyte of interest.Some exemplary hydrogel formulations are described in WO 97/02811,published Jan. 30, 1997. The ionically conductive material may comprisea biocide. For example, during manufacture of an autosensor assembly,one or more biocides may be incorporated into the ionically conductivematerial. Biocides of interest include, but are not limited to,compounds such as chlorinated hydrocarbons; organometallics; hydrogenreleasing compounds; metallic salts; organic sulfur compounds; phenoliccompounds (including, but not limited to, a variety of Nipa HardwickeInc. liquid preservatives registered under the trade names Nipastat®,Nipaguard®, Phenosept®, Phenonip®, Phenoxetol®, and Nipacide®);quaternary ammonium compounds; surfactants and other membrane-disruptingagents (including, but not limited to, undecylenic acid and its salts),combinations thereof, and the like.

[0085] The term “buffer” refers to one or more components which areadded to a composition in order to adjust or maintain the pH of thecomposition.

[0086] The term “electrolyte” refers to a component of the ionicallyconductive medium which allows an ionic current to flow within themedium. This component of the ionically conductive medium can be one ormore salts or buffer components, but is not limited to these materials.

[0087] The term “collection reservoir” is used to describe any suitablecontainment method or device for containing a sample extracted from abiological system. For example, the collection reservoir can be areceptacle containing a material which is ionically conductive (e.g.,water with ions therein), or alternatively it can be a material, such asa sponge-like material or hydrophilic polymer, used to keep the water inplace. Such collection reservoirs can be in the form of a hydrogel (forexample, in the shape of a disk or pad). Hydrogels are typicallyreferred to as “collection inserts.” Other suitable collectionreservoirs include, but are not limited to, tubes, vials, strips,capillary collection devices, cannulas, and miniaturized etched, ablatedor molded flow paths.

[0088] A “collection insert layer” is a layer of an assembly or laminatecomprising a collection reservoir (or collection insert) located, forexample, between a mask layer and a retaining layer.

[0089] A “laminate” refers to structures comprised of, at least, twobonded layers. The layers may be bonded by welding or through the use ofadhesives. Examples of welding include, but are not limited to, thefollowing: ultrasonic welding, heat bonding, and inductively coupledlocalized heating followed by localized flow. Examples of commonadhesives include, but are not limited to, chemical compounds such as,cyanoacrylate adhesives, and epoxies, as well as adhesives having suchphysical attributes as, but not limited to, the following: pressuresensitive adhesives, thermoset adhesives, contact adhesives, and heatsensitive adhesives.

[0090] A “collection assembly” refers to structures comprised of severallayers, where the assembly includes at least one collection insertlayer, for example a hydrogel. An example of a collection assembly asreferred to in the present invention is a mask layer, collection insertlayer, and a retaining layer where the layers are held in appropriatefunctional relationship to each other but are not necessarily a laminate(i.e., the layers may not be bonded together. The layers may, forexample, be held together by interlocking geometry or friction).

[0091] The term “mask layer” refers to a component of a collectionassembly that is substantially planar and typically contacts both thebiological system and the collection insert layer. See, for example,U.S. Pat. Nos. 5,735,273, 5,827,183, and 6,201,979, all hereinincorporated by reference.

[0092] The term “gel retaining layer” or “gel retainer” refers to acomponent of a collection assembly that is substantially planar andtypically contacts both the collection insert layer and the electrodeassembly.

[0093] The term “support tray” typically refers to a rigid,substantially planar platform and is used to support and/or align theelectrode assembly and the collection assembly. The support trayprovides one way of placing the electrode assembly and the collectionassembly into the sampling system.

[0094] An “autosensor assembly” refers to a structure generallycomprising a mask layer, collection insert layer, a gel retaining layer,an electrode assembly, and a support tray. The autosensor assembly mayalso include liners where the layers are held in approximate, functionalrelationship to each other. Exemplary collection assemblies andautosensor structures are described, for example, in InternationalPublication WO 99/58190, published Nov. 18, 1999; and U.S. Pat. Nos.5,735,273 and 5,827,183. The mask and retaining layers are preferablycomposed of materials that are substantially impermeable to the analyte(chemical signal) to be detected; however, the material can be permeableto other substances. By “substantially impermeable” is meant that thematerial reduces or eliminates chemical signal transport (e.g., bydiffusion). The material can allow for a low level of chemical signaltransport, with the proviso that chemical signal passing through thematerial does not cause significant edge effects at the sensingelectrode.

[0095] The terms “about” or “approximately” when associated with anumeric value refers to that numeric value plus or minus 10 units ofmeasure (i.e. percent, grams, degrees or volts), preferably plus orminus 5 units of measure, more preferably plus or minus 2 units ofmeasure, most preferably plus or minus 1 unit of measure.

[0096] By the term “printed” is meant a substantially uniform depositionof an electrode formulation onto one surface of a substrate (i.e., thebase support). It will be appreciated by those skilled in the art that avariety of techniques may be used to effect substantially uniformdeposition of a material onto a substrate, e.g., Gravure-type printing,extrusion coating, screen coating, spraying, painting, electroplating,laminating, or the like.

[0097] The term “physiological effect” encompasses effects produced inthe subject that achieve the intended purpose of a therapy. In preferredembodiments, a physiological effect means that the symptoms of thesubject being treated are prevented or alleviated. For example, aphysiological effect would be one that results in the prolongation ofsurvival in a patient.

[0098] “Parameter” refers to an arbitrary constant or variable soappearing in a mathematical expression that changing it give variouscases of the phenomenon represented (McGraw-Hill Dictionary ofScientific and Technical Terms, S. P. Parker, ed., Fifth Edition,McGraw-Hill Inc., 1994). In the context of the GlucoWatch® (Cygnus,Inc., Redwood City, Calif.) biographer, a parameter is a variable thatinfluences the value of the blood glucose level as calculated by analgorithm.

[0099] “Decay” refers to a gradual reduction in the magnitude of aquantity, for example, a current detected using a sensor electrode wherethe current is correlated to the concentration of a particular analyteand where the detected current gradually reduces but the concentrationof the analyte does not.

[0100] 2. General Overview of the Invention

[0101] Before describing the present invention in detail, it is to beunderstood that this invention is not limited to particular types ofmicroprocessors, monitoring systems, computational methods or processparameters, as use of such particulars may be selected in view of theteachings of the present specification. It is also to be understood thatthe terminology used herein is for the purpose of describing particularembodiments of the invention only, and is not intended to be limiting.

[0102] Although a number of methods and materials similar or equivalentto those described herein can be used in the practice of the presentinvention, the preferred materials and methods are described herein.

[0103] There are many methods of measuring an analyte that rely oncorrelation of a measured signal (e.g., an amperometric signal) that issubsequently related to analyte amount or concentration. Suchanalytically useful signals typically have kinetic and equilibriumcomponents. FIG. 2 shows two exemplary signals and their correspondingkinetic and equilibrium regions. In this regard, electrode sensorelements typically have kinetic and equilibrium regions of response overtime. For example, when plotting sensor response (i.e., generatedsignal) against time, there may be a period of rapidly changing signal(e.g., increasing or decreasing signal) corresponding to a kineticregion, followed by a plateau corresponding to an equilibrium region.Rate methods for determining an analyte concentration based on adetected signal can be rapid and modified to include certaincorrections, for example a background correction. However, suchtraditional rate methods have the following disadvantages: largevariable dependencies; limited linear ranges; high dependency on noise;and, low sensitivity. Rate methods, for example, can have a highvariability related to changes in the temperature at which the signal isbeing collected.

[0104] Equilibrium methods have certain advantages such as, smallvariable dependencies, extended linear ranges, and lower dependence onsignal noise. However, when using such equilibrium methods theaquisition of data is slow and typically requires background correctionas well. Further, in such methods there is a higher chance of sidereactions taking place given the long time frame required to obtain theequilibrium data.

[0105] End-point methods for determining the amount or concentration ofan analyte suffer from some of the same limitations as equilibriummethods, in particular, end-point methods require the essentiallycomplete reaction of all analyte present in a sample. Accordingly, asignal-producing reaction correlated to analyte amount or concentrationin a first sample must be completed before such a determination can bemade for a second sample. Further, end-point measurements aresusceptible to a higher chance of side reactions.

[0106] An alternative to equilibrium and end-point measurement methodsis to take a fixed-point measurement at some time point beforecompletion of the signal-producing reaction, thus providing a timesavings relative to the end-point method (as described above). The timepoint used in a fixed-point measurement is typically chosen empirically,based on the type of signal being generated. Such fixed pointmeasurements, however, are often prone to increased error due tovariables affecting the signal measurement (for example, temperature,pH, electrode sensitivity). Fixed-point measurements taken during a timeperiod when the signal is rapidly changing tend to be most error prone.However, even fixed time-point measurements taken during time periodswhere stable signals are typically produced can be affected byvariables, for example, background noise, or spikes or pulses in theelectrode response.

[0107] The present invention provides methods to reduce the timerequired for the measurement of analyte concentration or amount. Thepredictive-kinetic method of the present invention estimates theequilibrium or end-point response of a sensor (i.e., generated signal)at infinite time, i.e., effective completion of the reaction, and showsless dependence on the effects of measurement variables, such astemperature, pH, electrode sensitivity, etc. As described herein, theend-point response can be useful in several ways. In one aspect, theestimated end-point value provides an estimated final background value.In another aspect, the estimated end-point value allows more accurateestimation of analyte-related signal.

[0108] The methods of the present invention provide, for example, thefollowing advantages: (a) reduction in the time lag between analyteextraction and measurement, and (b) reduction in the sensitivity tochanges in measurement variables (e.g., reduced noise). In the method ofthe present invention, data from the transient, or kinetic, region of anintegrated signal are used with suitable models and curve-fittingmethods to predict the signal that would be measured for the system atthe completion of the reaction.

[0109] Following here is a general description of one embodiment of thepredictive-kinetic method of the present invention.

[0110] (i) A series of measurements of kinetic, measured data (e.g.,amperometric signal) is collected using a selected sampling system. Themeasured data typically takes the form of a response curve (e.g., FIG.2, curves A or B; in the figure—Curve B can represent the integratedform of curve A) with response measured relative to time. Based on theresults of the measured data (e.g., comprising a series of data pointsover time), or preliminary transformations of the series, such as,integration to obtain an “area under the curve”, a mathematical model isselected which describes the curve created by the series ofmeasurements. That is, a mathematical model is used to fit a curve tothe measured response curve. With reference to curve B of FIG. 2, such amathematical model may be, for example, an equation (Eqn.) defining apseudo-first-order reaction or process

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0111] where S_(o), S_(t), and S_(∞) are initial, intermediate, andfinal (or end-point) signals, k and t are the observed first-order rateconstant (also referred to herein as the pseudo-first order rateconstant) and time, respectively. In this example the value of theparameter S_(∞) is what is being determined, i.e., a predicted“end-point” analyte-related value for the signal-producing reaction.Similarly, this process can be applied to curve A where the S_(∞) valuemay correspond to a final background value.

[0112] (ii) the parameters for use in the model are identified andinitial values of the parameters, for example, S_(o), S_(∞), and k, areestimated.

[0113] (iii) these initial values are used to predict S_(t) at aselected number of time points

[0114] (iv) an error is determined, e.g., sum of (S_(t) measured minusS_(t) predicted)²

[0115] (v) the parameters are iteratively estimated until the errorbetween predicted and measured values falls within an acceptable range(e.g., using a chi-square test) or until no further significant changeis seen in the calculated error, at which time iterative estimation ofthe parameters is stopped. At this point the prediction of S_(∞), hasbeen optimized.

[0116] The steps described above can be repeated any number of times toobtain a series of measurements (e.g., at least twice, and preferablyfrequently repeated over a time period, for example, several times anhour over a 12 or 24 hour period).

[0117] The estimated signal corresponding to S_(∞)is then converted to acorresponding amount or concentration of the analyte of interest, forexample, where the predicted response curve is fitted to measured dataof the form presented in FIG. 2, curve A, by integration of thepredicted signal response curve and conversion of the end-point valueobtained by integration to an analyte specific value using a calibrationvalue.

[0118] Many models are useful in the practice of the predictive-kineticmethod of the present invention, including, but not limited to thefollowing: first order, second order, variable-order, parallel multiplefirst order, hyperbolic and first order, hyperbolic, linear Muller,Massart, Buck, one point fixed time, first and zero order, first andzero order with quadratic terms, first order and square root,first-order and square root with time shift, n^(th) order, consecutivefirst order, Michaelis-Menton, flux, flux with time shift, sigmoidal,and combinations thereof (see, for example, the formulae presented inFIG. 13). Such models may comprise zero order terms as well. Someanalysis methods relating to a flux model (Olsson, B., et al., Anal.Chem. 58:1046-1052, 1986) and a pseudoequilibrium model (Chen, W., etal., Analytica Chimica Acta 388:231-242, 1999) have been described fordifferent applications. Further, using standard mathematicalmanipulations empirical models can be established based on collecteddata sets.

[0119] Further, when the predictive-kinetic methods of the presentinvention are used to estimate a final background value, the finalbackground value can be used to provide a correction to the predictedresponse curve, for example, by background subtraction.

[0120] While not wishing to be bound by any particular theory as to whythe present invention works, the following mathematical description ofthe predictive-kinetic method of the present invention is provided tofurther general understanding of the invention. The time dependentresponse of a reaction or process may be modeled to fit a selectedsystem in a number of ways including a first-order model such as ispresented is Equation (Eqn.) 1:

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0121] where S_(o), S_(t), and S_(∞) are initial, intermediate, andfinal signals, k and t are the observed first-order rate constant (orpseudo-first order rate constant) and time, respectively. The rateconstant is considered a pseudo-first order rate constant because itaccounts for the reactions and processes that are contained in themeasured responses. The simultaneous estimation of S₀, S_(∞) and k issimplified by use of a linear model rather than the non-linear model inEquation 1 (Mieling, G. E., and Pardue, H. L., Anal. Chemistry 50(1978)1611-1618). An expansion of Equation 1 as functions of S_(o), S_(∞), andk using a simplified Taylor series gives an equation that is linear inparameter increments as follows: $\begin{matrix}{S_{t} = {S_{t}^{o} + {\frac{\partial S_{t}^{o}}{\partial k}\delta \quad k} + {\frac{\partial S_{t}^{o}}{\partial S_{o}}\delta \quad S_{o}} + {\frac{\partial S_{t}^{o}}{\partial S_{\infty}}\delta \quad S_{\infty}}}} & \text{(Eqn.~~2)}\end{matrix}$

[0122] where S_(t) ^(o) is an initial estimate of S_(t) expressed interms of initial estimates of S_(o), S_(∞), and k. The partialderivatives are also derived from initial estimates of S_(o), S_(∞), andk inserted into the following expressions: $\begin{matrix}{\frac{\partial S_{t}^{o}}{\partial k} = {t\left( {S_{\infty} - S_{o}} \right)}} & \text{(Eqn.~~3a)}\end{matrix}$

$\begin{matrix}{\frac{\partial S_{t}^{o}}{\partial S_{o}} = ^{- {kt}}} & \text{(Eqn.~~3b)}\end{matrix}$

[0123] and $\begin{matrix}{\frac{\partial S_{t}^{o}}{\partial S_{\infty}} = {1 - ^{- {kt}}}} & \text{(Eqn.~~3c)}\end{matrix}$

[0124] A minimization protocol, such as a multiple-linear regressionprogram (e.g., Levenberg-Marquardt Method or simplex optimization) isused to evaluate values of δk, δS_(o) and δS_(∞) which when added toinitial estimate of k, S_(o) and S_(∞) will give values of k, S_(o) andS_(∞) which represent the best fit of the experimental data to thelinearized first-order equation.

[0125] The criterion used to obtain the best fit involves theminimization of the following function: $\begin{matrix}{\chi^{2} = {\frac{1}{S_{sd}^{2}}{\sum\left\{ {S_{t} - S_{t}^{o} + {\frac{\partial S_{t}^{o}}{\partial k}\delta \quad k} + {\frac{\partial S_{t}^{o}}{\partial S_{o}}\delta \quad S_{o}} + {\frac{\partial S_{t}^{o}}{\partial S_{\infty}}\delta \quad S_{\infty}}} \right\}}}} & \text{(Eqn.~~4)}\end{matrix}$

[0126] where χ² is chi-square, and S_(sd) is the estimated standarddeviation of the signal measurements. The function is minimized bysetting the first derivative of χ² with respect to δS_(o),δS_(∞) and δkequal to zero and solving the normal equations that result for theparameters δS_(o), δS_(∞) and δk.

[0127] Initial estimates of S_(o), S_(∞) and k, are usually in error;truncated Taylor's series in Equation 2 is only an approximation, of thenonlinear model. However, successive iterations of the proceduredescribed above are used to obtain best estimates of these parameters.Iterations are continued until the change in χ² is less than 0.05%.Typically, this requires 3 to 5 iterations (Mieling, G. E., and Pardue,H. L., Anal. Chemistry 50(1978) 1611-1618).

[0128] Initial estimates of S_(o), S_(∞), and k, can be performed, forexample, using three optional approaches: Manual, user defined values;Cornell Partial Sums; or successive integration.

[0129] Slow convergence to minimum chi-square occurs where thehypersurface does not approximate a paraboloid. The Marquardt algorithmcan be used to sense this condition and a procedure approximating themethod of steepest descent is used to approach the approximate parabolichypersurface where the regression method takes over and proceeds asdescribed above. When the process is completed, the projected change insignal may be computed. For the above-described first order model, theprojected change in signal may be represented as follows:

ΔS=S _(∞) −S _(o)  (Eqn. 5)

[0130] Following the method of the present invention, to estimateequilibrium or end-point signal for any time-dependent response notmonitored to completion, multipoint data from the transient region areused with suitable models and curve-fitting methods to predict thesignal that would be measured for the system at equilibrium or atcompletion of the reaction. This approach was illustrated above for aprocess that follows apparent first-order kinetics. A multiple-linearregression program is used to compute initial and equilibrium values ofthe signal and the first-order rate constant that represent the “bestfit” of the signal vs. time data to a first-order model. Analyteconcentration or amount is computed from the difference between initialand final signal values.

[0131] Although not wishing to be bound by a particular theory, thefollowing explanation is provided to assist in understanding the methodof the present invention. The method derives its reduced dependency uponmeasurement variables, at least in part, from the fact that the totalchange in signal at equilibrium is less dependent upon the variablesthan are the kinetic data, and the rate constant used to define thefirst-order process is determined independently for each sample whilethe analysis is in progress. Simply stated, while the reaction rate ishighly variable, the extent is relatively less so. Because the methodcomputes the signal change that would be measured if the responses weremonitored to completion, the method should have characteristics moreclosely related to the equilibrium methods than to conventional kineticmethods provided the first-order rate constant used to fit the model isthe correct one for conditions existing for each individual sample. Themultiple regression method satisfies this criterion by determining thevalue of the rate constant, as well as the initial and final signalsthat represents the “best fit” to the data for each individual sample.

[0132] In the general method of the invention, a measured signal isobtained from a sensing device, which signal is related to a targetanalyte present in the biological system. The measured signal can beobtained using any suitable sensing methodology including, for example,methods which rely on direct contact of a sensing apparatus with asystem; methods which extract samples from the system by invasive,minimally invasive, and non-invasive sampling techniques, wherein thesensing apparatus is contacted with the extracted sample; methods whichrely on indirect contact of a sensing apparatus with the system; and thelike. In preferred embodiments of the invention, methods are used toextract samples from a biological system using minimally invasive ornon-invasive sampling techniques. The sensing apparatus used with any ofthe above-noted methods can employ any suitable sensing element toprovide the signal including, but not limited to, physical, chemical,electrochemical, or like elements. In preferred embodiments of theinvention, a biosensor is used which comprises an electrochemicalsensing element.

[0133] The measured signal obtained using any of the above describedmethodologies is then converted into an analyte specific value of knownunits to provide an interpretation of the signal obtained from thesensing device. The interpretation uses a mathematical transformation tomodel the relationship between a measured response in the sensing deviceand a corresponding analyte-specific value (in the present invention, apredictive-kinetic method). Thus, a calibration step is used herein torelate, for example, an electrochemical signal (detected by a biosensor)with the concentration of a target analyte in a biological system.

[0134] The predicted analyte values can optionally be used in asubsequent step to control an aspect of the biological system. In oneembodiment, predicted analyte values are used to determine when, and atwhat level, a constituent should be added to the biological system inorder to control an aspect of the biological system. In a preferredembodiment, the analyte value can be used in a feedback control loop tocontrol a physiological effect in the biological system.

[0135] The present invention includes, but is not limited to, methods,devices, algorithms, computer programs, equations, statistical methods,processes, and microprocessors, for use singly or in combination formeasuring an analyte as described herein by the present invention. Inone aspect, the present invention describes a method for measuring ananalyte present in a subject. The analyte may, for example, be extractedfrom the subject transdermally using a sampling system that is inoperative contact with a skin or mucosal surface of the subject. Fromthis extracted sample a measured signal is obtained (e.g., using asensing device) where the measured signal comprises a measured signalresponse curve derived from the extracted analyte, wherein the measuredsignal is specifically related to the amount or concentration ofanalyte, and the measured signal response curve comprises kinetic andequilibrium regions. In order to predict an analyte end-point value, amathematical model comprising selected parameters is used, wherein themodel describes the measured signal response curve. Numerous exemplary,suitable models are described herein (see, for example, FIG. 13).Further an error minimization method is typically employed. Theparameters are iteratively estimated using the model and errorminimization method to provide a predicted response curve correspondingto the measured signal response curve, wherein (i) the errorminimization method provides a calculated error based on differencesbetween the predicted and measured signal response curves, and (ii) theestimating is iteratively performed until the calculated error betweenthe predicted and measured signal response curves falls within anacceptable range or until no further statistically significant change isseen in the calculated error. At that time iterative estimation of theparameters is stopped. The iterative estimation and error minimizationresults in a predicted response curve corresponding to the measuredsignal response curve, the predicted response curve yields a predictedend-point value. This predicted end-point value may correspond to abackground value (e.g., FIG. 2, curve A) remaining after analytespecific signal is depleted (i.e., a final background value) or theend-point value may provide an analyte-related measurement (e.g., FIG.2, curve B) correlated to the amount or concentration of the analyte(i.e., an end-point analyte-related value obtained, for example, byintegration of the predicted response curve). An end-pointanalyte-related value may be further manipulated to give the amount orconcentration of analyte by, for example, performing a calibration step.

[0136] In one embodiment, the present invention includes one or moremicroprocessors programmed to control a measurement cycle (i.e.,programmed to control sampling and sensing devices) and to execute thecomputations of the predictive-kinetic methods described herein. Suchmicroprocessors are useful devices alone (e.g., as a durable componentof a device where the sampling and sensing devices are disposable and/orreplaceable) and when placed in combination with further components(e.g., as a complete unit comprising such one or more microprocessors, asampling device, a sensing device, and associated components such asanalyte display screens, warning alert generators, power supply, etc.).

[0137] In a further embodiment, the present invention includes amonitoring system (in combination, as well as in the embodiments of itsseparate components) for frequent measurement of an analyte amount orconcentration present in a subject. The following components of thesystem are in operative combination/communication:

[0138] (A) a sampling device for frequently extracting the analyte fromthe subject (for example, a sampling device that is adapted forextracting the analyte across a skin or mucosal surface of the subjector, in an alternative embodiment, a subcutaneous sampling device);

[0139] (B) a sensing device in operative contact with the analyteextracted by the sampling device, wherein the sensing device obtains ameasured signal, comprising a measured signal response curve, from theextracted analyte, wherein the measured signal is specifically relatedto the amount or concentration of analyte, and the measured signalresponse curve comprises kinetic and equilibrium regions; and

[0140] (C) one or more microprocessors capable of being in operativecommunication with the sampling device and the sensing device. Themicroprocessor is capable of controlling the sampling device and thesensing device to obtain a series of measured signals in the form ofresponse curves at selected time intervals during a measurement period.Further, the microprocessor is capable of predicting measurement valuesfor each measured signal in the series by employing (i) a mathematicalmodel comprising selected parameters, wherein the model describes themeasured signal response curve, and (ii) an error minimization method,to iteratively estimate values of the parameters using the model anderror minimization method to provide a predicted response curvecorresponding to the measured signal response curve. The errorminimization method provides a calculated error based on differencesbetween the predicted and measured signal response curves. Theestimating is iteratively performed until the calculated error betweenthe predicted and measured signal response curves falls within anacceptable range or until no further statistically significant change isseen in the calculated error, at which time iterative estimation of theparameters is stopped. This iterative estimation and error minimizationresults in a predicted response curve corresponding to the measuredsignal response curve and the predicted response curve yields apredicted end-point value. In some embodiments, each predicted end-pointanalyte-related value of the series may be correlated with a measurementvalue indicative of the amount or concentration of analyte present inthe subject.

[0141] The above general methods and devices can, of course, be usedwith a wide variety of detection systems, target analytes, and/orsensing techniques. The determination of particularly suitablecombinations is within the skill of the ordinarily skilled artisan whendirected by the present disclosure. Although these methods are broadlyapplicable to measuring any chemical analyte and/or substance in asystem, the invention is expressly exemplified for use in a transdermalsampling system which uses an electrochemical biosensor to quantify orqualify glucose or a glucose metabolite.

[0142] 3. Exemplary Sampling Systems

[0143] An automatic sampling system may be used to monitor levels ofanalyte. One such exemplary sampling system is described herein formonitoring glucose levels in a biological system via the transdermallyextraction of the analyte (e.g., glucose) from the biological system,particularly an animal subject. Transdermal extraction is carried out byapplying an electrical current or ultrasonic radiation to a tissuesurface at a collection site. The electrical current is used to extractsmall amounts of glucose from the subject into a collection reservoir.The collection reservoir is in contact with a sensor element (biosensor)which provides for measurement of glucose concentration in the subject.As glucose is transdermally extracted into the collection reservoir, theanalyte reacts with the glucose oxidase within the reservoir to producehydrogen peroxide. The presence of hydrogen peroxide generates a currentat the biosensor electrode that is directly proportional to the amountof hydrogen peroxide in the reservoir. This current provides a signalwhich can be detected and interpreted (for example, employing thepredictive-kinetic method described herein) by an associated systemcontroller to provide a glucose concentration value or amount fordisplay.

[0144] In the use of the sampling system, a collection reservoir iscontacted with a tissue surface, for example, on the stratum corneum ofa subject's skin. An electrical current is then applied to the tissuesurface in order to extract glucose from the tissue into the collectionreservoir. Extraction is carried out, for example, frequently over aselected period of time. The collection reservoir is analyzed, at leastperiodically and typically frequently, to measure glucose concentrationtherein. The measured value correlates with the subject's blood glucoselevel.

[0145] To sample the analyte, one or more collection reservoirs areplaced in contact with a tissue surface on a subject. The ionicallyconductive material within the collection reservoir is also in contactwith an electrode (for reverse iontophoretic extraction) which generatesa current sufficient to extract glucose from the tissue into thecollection reservoir. Referring to FIG. 1, an exploded view of exemplarycomponents comprising one embodiment of an autosensor for use in aniontophoretic sampling system is presented. The autosensor componentsinclude two biosensor/iontophoretic electrode assemblies, 104 and 106,each of which have an annular iontophoretic electrode, respectivelyindicated at 108 and 110, which encircles a biosensor electrode 112 and114. The electrode assemblies 104 and 106 are printed onto a polymericsubstrate 116 which is maintained within a sensor tray 118. A collectionreservoir assembly 120 is arranged over the electrode assemblies,wherein the collection reservoir assembly comprises two hydrogel inserts122 and 124 retained by a gel retaining layer 126 and mask layer 128.Further release liners may be included in the assembly, for example, apatient liner 130, and a plow-fold liner 132. In an alternativeembodiment, the electrode assemblies can include bimodal electrodes. Apolyurethane mask layer 128 as described in PCT Publication No. WO97/10356, published Mar. 20, 1997, may be present. Other embodiments ofthe autosensor are described in WO 99/58190, published Nov. 18, 1999.

[0146] The mask and retaining layers are preferably composed ofmaterials that are substantially impermeable to the analyte (e.g.,glucose) to be detected (see, for example, U.S. Pat. Nos. 5,735,273, and5,827,183). By “substantially impermeable” is meant that the materialreduces or eliminates analyte transport (e.g., by diffusion). Thematerial can allow for a low level of analyte transport, with theproviso that the analyte that passes through the material does not causesignificant edge effects at the sensing electrode used in conjunctionwith the mask and retaining layers. Examples of materials that can beused to form the layers include, but are not limited to, polyester,polyester derivatives, other polyester-like materials, polyurethane,polyurethane derivatives and other polyurethane-like materials.

[0147] The components shown in exploded view in FIG. 1 are intended foruse in a automatic sampling system which is configured to be worn likean ordinary wristwatch, as described in PCT Publication No. WO 96/00110,published Jan. 4, 1996. The wristwatch housing can further includesuitable electronics (e.g., one or more microprocessor(s), memory,display and other circuit components) and power sources for operatingthe automatic sampling system. The one or more microprocessors maycontrol a variety of functions, including, but not limited to, controlof a sampling device, a sensing device, aspects of the measurement cycle(for example, timing of sampling and sensing, and alternating polaritybetween electrodes), connectivity, computational methods, differentaspects of data manipulation (for example, acquisition, recording,recalling, comparing, and reporting), etc.

[0148] The sensing electrode can be, for example, a Pt-comprisingelectrode configured to provide a geometric surface area of about 0.1 to3 cm², preferably about 0.5 to 2 cm², and more preferably about 1 cm².This particular configuration is scaled in proportion to the collectionarea of the collection reservoir used in the sampling system of thepresent invention, throughout which the extracted analyte and/or itsreaction products will be present. The electrode composition isformulated using analytical- or electronic-grade reagents and solventswhich ensure that electrochemical and/or other residual contaminants areavoided in the final composition, significantly reducing the backgroundnoise inherent in the resultant electrode. In particular, the reagentsand solvents used in the formulation of the electrode are selected so asto be substantially free of electrochemically active contaminants (e.g.,anti-oxidants), and the solvents in particular are selected for highvolatility in order to reduce washing and cure times. Some electrodeembodiments are described in European Patent Publication 0 942 278 A2,published Sep. 15, 1999.

[0149] The reactive surface of the sensing electrode can be comprised ofany electrically conductive material such as, but not limited to,platinum-group metals (including, platinum, palladium, rhodium,ruthenium, osmium, and iridium), nickel, copper, silver, and carbon, aswell as, oxides, dioxides, combinations or alloys thereof. Somecatalytic materials, membranes, and fabrication technologies suitablefor the construction of amperometric biosensors were described byNewman, J. D., et al. (Analytical Chemistry 67(24), 4594-4599, 1995).

[0150] Any suitable iontophoretic electrode system can be employed, anexemplary system uses a silver/silver chloride (Ag/AgCl) electrodesystem. The iontophoretic electrodes are formulated typically using twoperformance criteria: (1) the electrodes are capable of operation forextended periods, preferably periods of up to 24 hours or longer; and(2) the electrodes are formulated to have high electrochemical purity inorder to operate within the present system which requires extremely lowbackground noise levels. The electrodes must also be capable of passinga large amount of charge over the life of the electrodes. With regard tooperation for extended periods of time, Ag/AgCl electrodes are capableof repeatedly forming a reversible couple which operates withoutunwanted electrochemical side reactions (which could give rise tochanges in pH, and liberation of hydrogen and oxygen due to waterhydrolysis). The Ag/AgCl electrode is thus formulated to withstandrepeated cycles of current passage in the range of about 0.01 to 1.0 mAper cm² of electrode area. With regard to high electrochemical purity,the Ag/AgCl components are dispersed within a suitable polymer binder toprovide an electrode composition which is not susceptible to attack(e.g., plasticization) by components in the collection reservoir, e.g.,the hydrogel composition. The electrode compositions are also typicallyformulated using analytical- or electronic-grade reagents and solvents,and the polymer binder composition is selected to be free ofelectrochemically active contaminants which could diffuse to thebiosensor to produce a background current.

[0151] The automatic sampling system can transdermally extract thesample over the course of a selected period of time using reverseiontophoresis. The collection reservoir comprises an ionicallyconductive medium, preferably the hydrogel medium described hereinabove.A first iontophoresis electrode is contacted with the collectionreservoir (which is typically in contact with a target, subject tissuesurface), and a second iontophoresis electrode is contacted with eithera second collection reservoir in contact with the tissue surface, orsome other ionically conductive medium in contact with the tissue. Apower source provides an electrical potential between the two electrodesto perform reverse iontophoresis in a manner known in the art. Asdiscussed above, the biosensor selected to detect the presence, andpossibly the level, of the target analyte (for example, glucose) withina reservoir is also in contact with the reservoir.

[0152] In practice, an electric potential (either direct current or amore complex waveform) is applied between the two iontophoresiselectrodes such that current flows from the first electrode through thefirst conductive medium into the skin, and back out from the skinthrough the second conductive medium to the second electrode. Thiscurrent flow extracts substances through the skin into the one or morecollection reservoirs through the process of reverse iontophoresis orelectroosmosis. The electric potential may be applied as described inPCT Publication No. WO 96/00110, published Jan. 4, 1996.

[0153] As an example, to extract glucose, the applied electrical currentdensity on the skin or tissue can be in the range of about 0.01 to about2 mA/cm². In order to facilitate the extraction of glucose, electricalenergy can be applied to the electrodes, and the polarity of theelectrodes can be, for example, alternated so that each electrode isalternately a cathode or an anode. The polarity switching can be manualor automatic. A device and method for sampling of substances usingalternating polarity is described in U.S. Pat. No. 5,771,890, issuedJun. 30, 1998.

[0154] When a bimodal electrode is used (e.g., U.S. Pat. No. 5,954,685,issued Sep. 21, 1999), during the reverse iontophoretic phase, a powersource provides a current flow to the first bimodal electrode tofacilitate the extraction of the chemical signal into the reservoir.During the sensing phase, a separate power source is used to providevoltage to the first sensing electrode to drive the conversion ofchemical signal retained in reservoir to electrical signal at thecatalytic face of the sensing electrode. The separate power source alsomaintains a fixed potential at the electrode where, for example hydrogenperoxide is converted to molecular oxygen, hydrogen ions, and electrons,which is compared with the potential of the reference electrode duringthe sensing phase. While one sensing electrode is operating in thesensing mode it is electrically connected to the adjacent bimodalelectrode which acts as a counter electrode at which electrons generatedat the sensing electrode are consumed.

[0155] The electrode subassembly can be operated by electricallyconnecting the bimodal electrodes such that each electrode is capable offunctioning as both an iontophoretic electrode and counter electrodealong with appropriate sensing electrode(s) and reference electrode(s).

[0156] A potentiostat is an electrical circuit used in electrochemicalmeasurements in three electrode electrochemical cells. A potential isapplied between the reference electrode and the sensing electrode. Thecurrent generated at the sensing electrode flows through circuitry tothe counter electrode (i.e., no current flows through the referenceelectrode to alter its equilibrium potential). Two independentpotentiostat circuits can be used to operate the two biosensors. For thepurpose of the present invention, the electrical current measured at thesensing electrode subassembly is the current that is correlated with anamount of chemical signal corresponding to the analyte.

[0157] The detected current can be correlated with the subject's bloodglucose concentration (using, for example, the predictive-kinetic methoddescribed herein) so that the system controller may display thesubject's actual blood glucose concentration as measured by the samplingsystem. Such statistical techniques can be formulated as algorithm(s)and incorporated in one or more microprocessor(s) associated with thesampling system.

[0158] In a further aspect of the present invention, thesampling/sensing mechanism and user interface may be found on separatecomponents. Thus, the monitoring system can comprise at least twocomponents, in which a first component comprises sampling mechanism andsensing mechanism that are used to extract and detect an analyte, forexample, glucose, and a second component that receives the analyte datafrom the first component, conducts data processing on the analyte datato determine an analyte concentration and then displays the analyteconcentration data. Typically, microprocessor functions (e.g., controlof a sampling device, a sensing device, aspects of the measurementcycle, computational methods, different aspects of data manipulation orrecording, etc.) are found in both components. Alternatively,microprocessing components may be located in one or the other of the atleast two components. The second component of the monitoring system canassume many forms, including, but not limited to, the following: awatch, a credit card-shaped device (e.g., a “smart card” or “universalcard” having a built-in microprocessor as described for example in U.S.Pat. No. 5,892,661), a pager-like device, cell phone-like device, orother such device that communicates information to the user visually,audibly, or kinesthetically.

[0159] Further, additional components may be added to the system, forexample, a third component comprising a display of analyte values or analarm related to analyte concentration, may be employed. In certainembodiments, a delivery unit is included in the system. An exemplarydelivery unit is an insulin delivery unit. Insulin delivery units, bothimplantable and external, are known in the art and described, forexample, in U.S. Pat. Nos. 5,995,860; 5,112,614 and 5,062,841.Preferably, when included as a component of the present invention, thedelivery unit is in communication (e.g., wire-like or wirelesscommunication) with the extracting and/or sensing mechanism such thatthe sensing mechanism can control the insulin pump and regulate deliveryof a suitable amount of insulin to the subject.

[0160] Advantages of separating the first component (e.g., including thebiosensor and iontophoresis functions) from the second component (e.g.,including some microprocessor and display functions) include greaterflexibility, discretion, privacy and convenience to the user. Having asmall and lightweight measurement unit allows placement of the twocomponents of the system on a wider range of body sites, for example,the first component may be placed on the abdomen or upper arm. Thiswider range of placement options may improve the accuracy throughoptimal extraction site selection (e.g., torso rather than extremities)and greater temperature stability (e.g., via the insulating effects ofclothing). Thus, the collection and sensing assembly will be able to beplaced on a greater range of body sites. Similarly, a smaller and lessobtrusive microprocessor and display unit (the second component)provides a convenient and discrete system by which to monitor analytes.The biosensor readouts and control signals will be relayed via wire-likeor wireless technology between the collection and sensing assembly andthe display unit which could take the form of a small watch, a pager, ora credit card-sized device. This system also provides the ability torelay an alert message or signal during nighttime use, for example, to asite remote from the subject being monitored.

[0161] In one embodiment, the two components of the device can be inoperative communication via a wire or cable-like connection. Operativecommunications between the components can be wireless link, i.e.provided by a “virtual cable,” for example, a telemetry link. Thiswireless link can be uni- or bi-directional between the two components.In the case of more than two components, links can be a combination ofwire-like and wireless.

[0162] 4. Exemplary Analytes

[0163] The analyte can be any specific substance, component, orcombinations thereof that one is desirous of detecting and/or measuringin a chemical, physical, enzymatic, or optical analysis. Thepredictive-kinetic method of the present invention may be employed aslong as the detection/measurement of the analyte is time dependent,e.g., the detection measurement method provides a response curve havinga kinetic region.

[0164] Analytes that can be measured using the methods of the presentinvention include, but are not limited to, amino acids, enzymesubstrates or products indicating a disease state or condition, othermarkers of disease states or conditions, drugs of abuse (e.g., ethanol,cocaine), therapeutic and/or pharmacologic agents (e.g., theophylline,anti-HIV drugs, lithium, anti-epileptic drugs, cyclosporin,chemotherapeutics), electrolytes, physiological analytes of interest(e.g., urate/uric acid, carbonate, calcium, potassium, sodium, chloride,bicarbonate (CO₂), glucose, urea (blood urea nitrogen), lactate and/orlactic acid, hydroxybutyrate, cholesterol, triglycerides, creatine,creatinine, insulin, hematocrit, and hemoglobin), blood gases (carbondioxide, oxygen, pH), lipids, heavy metals (e.g., lead, copper), and thelike. Analytes in non-biological systems may also be evaluated using themethods of the present invention.

[0165] In preferred embodiments, the analyte is a physiological analyteof interest, for example glucose, or a chemical that has a physiologicalaction, for example a drug or pharmacological agent.

[0166] In order to facilitate detection of the analyte, an enzyme (orenzymes) can be disposed within the one or more collection reservoirs.The selected enzyme is capable of catalyzing a reaction with theextracted analyte to the extent that a product of this reaction can besensed, e.g., can be detected electrochemically from the generation of acurrent which current is detectable and proportional to the amount ofthe analyte which is reacted. In one embodiment of the presentinvention, a suitable enzyme is glucose oxidase, which oxidizes glucoseto gluconic acid and hydrogen peroxide. The subsequent detection ofhydrogen peroxide on an appropriate biosensor electrode generates twoelectrons per hydrogen peroxide molecule creating a current that can bedetected and related to the amount of glucose entering the device.Glucose oxidase (GOx) is readily available commercially and has wellknown catalytic characteristics. However, other enzymes can also be usedsingly (for detection of individual analytes) or together (for detectionof multiple analytes), as long as they specifically catalyze a reactionwith an analyte or substance of interest to generate a detectableproduct in proportion to the amount of analyte so reacted.

[0167] In like manner, a number of other analyte-specific enzyme systemscan be used in the invention, which enzyme systems operate on much thesame general techniques. For example, a biosensor electrode that detectshydrogen peroxide can be used to detect ethanol using an alcohol oxidaseenzyme system, or similarly uric acid with urate oxidase system,cholesterol with a cholesterol oxidase system, and theophylline with axanthine oxidase system.

[0168] In addition, the oxidase enzyme (used for hydrogenperoxidase-based detection) can be replaced or complemented with anotherredox system, for example, the dehydrogenase-enzyme NAD-NADH, whichoffers a separate route to detecting additional analytes.Dehydrogenase-based sensors can use working electrodes made of gold orcarbon (via mediated chemistry). Examples of analytes suitable for thistype of monitoring include, but are not limited to, cholesterol,ethanol, hydroxybutyrate, phenylalanine, triglycerides, and urea.

[0169] Further, the enzyme can be eliminated and detection can rely ondirect electrochemical or potentiometric detection of an analyte. Suchanalytes include, without limitation, heavy metals (e.g., cobalt, iron,lead, nickel, zinc), oxygen, carbonate/carbon dioxide, chloride,fluoride, lithium, pH, potassium, sodium, and urea. Also, the samplingsystem described herein can be used for therapeutic drug monitoring, forexample, monitoring anti-epileptic drugs (e.g., phenytoin), chemotherapy(e.g., adriamycin), hyperactivity (e.g., ritalin), andanti-organ-rejection (e.g., cyclosporin).

[0170] Preferably, a sensor electrode is able to detect the analyte thathas been extracted into the one or more collection reservoirs whenpresent at nominal concentration levels. Suitable exemplary biosensorelectrodes and associated sampling systems as described in are describedin PCT Publication Nos. WO 97/10499, published Mar. 20, 1997 and WO98/42252, published Oct. 1, 1998.

[0171] Further, the predictive-kinetic methods of the present inventionfacilitate analysis of multiple analytes obtained in a single sample(e.g., a sample collected into a single reservoir using transdermalextraction), even when such multiple analytes are being detected by acommon reaction product. For example, a sensing device may be used thatemploys several oxidase enzymes, e.g., lactate oxidase, uricase, andglucose oxidase. Each of these enzymes has the ability to generatehydrogen peroxide when contacted by their respective substrates. Asingle sensor sensitive to, for example, hydrogen peroxide (e.g., aplatinum electrode), cannot differentiate between peroxide originatingfrom glucose, uric acid or lactic acid. However, by employing thepredictive-kinetic methods of the present invention, the apparent rateconstant for each reaction and the concentration of each analyte can beresolved, that is, the predictive-kinetic method can resolve theindividual contributions to overall, final, peroxide-mediated signal.Thus, with suitable computing power, the concentrations of each analytecan be obtained. Variables, such as, pH and enzyme concentration, allowmanipulation of the apparent rate constants of each enzyme to aidresolution and minimize interference between components. Further, asystem of weighting factors could be employed as well, where, forexample, contributions by different components are weighted differentlybased on their known contribution to overall signal.

[0172] Typically, the reactions with substrate to form detectableproduct, as facilitated by different enzymes, do not interfere with oneanother. The predictive-kinetic methods described herein areparticularly useful for detection of multiple analytes using a commonreaction product, for example, hydrogen peroxide, when there are atleast three-fold differences, preferably five- to ten-fold difference orhigher, in the reaction rate constants for conversion of the differentanalytes to the common reaction product. For example, detection ofglucose and urea in a single sample may be facilitated by the use of theenzymes glucose oxidase and uricase (urate oxidase) both of which yieldhydrogen peroxide as the common, detectable reaction product. The k_(m)of glucose oxidase is approximately 3.3×10⁻² molar and the km of uricaseis approximately 10⁻⁵ molar. For example, signals corresponding toglucose and urea can be resolved within a single signal response curvebased on the apparent rate constants (i.e., the k_(m)) of the tworeactions using the parallel first order predictive-kinetic modeldescribed herein.

[0173] In the example described above a common reaction product isformed (i.e., hydrogen peroxide); however, this is not a requirement. Asingle sensor may detect multiple analytes and/or reaction products ofanalytes. For example, a platinum sensor could be used to detecttyrosine and glucose in a single sample. The tyrosine is detected, forexample, by direct electrochemical oxidation at a suitable electrodepotential (e.g., approximately 0.6V vs. Ag/AgCl). The glucose isdetected, e.g., using glucose oxidase and detecting the hydrogenperoxide reaction product. For example, signals corresponding totyrosine and glucose can be resolved within a single signal responsecurve based on the apparent rate constants (i.e., the km) of the tworeactions using the parallel first order predictive-kinetic modeldescribed herein.

[0174] Generally when detecting multiple analytes with a single sensorit is preferred that, within a single response curve, the primarysignals corresponding to each analyte are separated in time, e.g., oneanalyte's reaction with the sensor is rapid (k₁) and a second analyte'sreaction with the sensor is slower (k₂), i.e., k₁>>k₂.

[0175] Different sensing devices and/or sensing systems can be employedas well to distinguish between signals. For example, a first gelcontaining glucose oxidase associated with a first platinum sensor canbe used for the detection of glucose, while a second gel containinguricase associated with a second platinum sensor can be used for thedetection of urea. The predictive-kinetic methods of the presentinvention may then used to individually model the signal response curvesgenerated at each sensor.

[0176] 5. Employing the Predictive-Kinetic Method in Glucose Measurement

[0177] A. Predictive-Kinetic Models

[0178] The GlucoWatch biographer is a device that provides frequent andautomatic glucose measurements. Glucose is extracted through the skinvia electro-osmosis and measured with an amperometric biosensor. Glucoseis extracted into a hydrogel of, for example, 0.18 mm in thickness,containing the enzyme, glucose oxidase. The enzyme converts theextracted glucose to hydrogen peroxide. The hydrogen peroxide isdetected by a Pt/C electrode composite directly under the hydrogel. Apotentiostat used to apply the polarizing voltage and collect theresulting current is part of the wearable device that displays values ofthe measured glucose to the user.

[0179] One variable affecting the measurement objective of thebiographer is the rate of mutarotation of the α to β forms of glucose(Kurnik R. T., et al., Journal of the Electrochemical Society 145 (1998)4119-4125). One goal of the present design of the biographer is that, inthe presence of excess enzyme loading and for a given extracted glucoseconcentration, the same response is measured by the biographerregardless of changes in the measurement variables. However, because themutarotation constant is dependent on changes in other measurementvariables, especially temperature, a long measurement time is requiredto ensure complete consumption of the glucose in the hydrogel (Pardue,H. L., Kinetic Aspects of Analytical Chemistry, Anal. Chim. Acta 69(1989) 216; Mieling, G. E., and Pardue, H. L., Anal. Chemistry 50(1978)1611-1618.). Accordingly, some of the measurement variables that canaffect the measurement objective of the biographer include mutarotation,diffusion (e.g., slow diffusion of glucose through the hydrogel), andelectrode kinetics. The latter results in apparent drift in sensorsensitivity. Use of the predictive-kinetic method of the presentinvention in combination with the biographer measurements provides atechnique that estimates the equilibrium or end-point responses of thebiographer. Further, the predictive-kinetic method of present inventionreduces the effects of measurement variables, such as temperature.

[0180] As described above, glucose (in a hydrogel) is converted tohydrogen peroxide, a current (in the order of nanoamps) is generated anddetected over time (typically resulting in a curve that looks like curveA in FIG. 2). The current is typically integrated to provide a curve ofnanocoloumb values (nC) relative to time. The resulting data can beschematically represented by curve B of FIG. 2.

[0181] To employ the predictive-kinetic method of the present invention,data from the kinetic region of the curve can be fitted to a model. Forbenchtop studies, described below, a first-order reaction or processfits the data well:

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0182] where S_(o), S_(t), and S_(∞)are initial, intermediate, and finalsignals, k and t are the observed first-order rate constant and time,respectively.

[0183] In one embodiment of the present invention, the method is used topredict an S_(∞) that corresponds to a final background value (e.g.,FIG. 2, curve A). This final background value can be used to provide abackground correction, for example, by using the final background valuefor background subtraction. In a further aspect, the method is used topredict an S_(∞) which corresponds to an analyte-related value, forexample, after integration of the predicted signal response curve basedon current, the end-point value corresponds to an equilibrium chargevalue (e.g., nC) which may then be further manipulated or directlyconverted to an amount or concentration of glucose. Accordingly, toobtain the predicted value of S_(∞), estimation of three values isrequired, S_(o), S_(∞), and k. Estimates of the values are made byiteratively using initial estimated values until, via an errorminimization method, the predicted curve matches the curve of the real(measured) data to within a predecided margin of error. The iterativeprocess is repeated until error minimization is achieved.

[0184] Experimental results from benchtop studies using the biographerare described in the Experimental section below. In these studies, thebiosensor was assembled and preconditioned for one hour. Ten microlitersof glucose solution of known concentration was then deposited on thehydrogel. The electrode response to the glucose was monitored. Thecurrent responses were then transferred to a computer for dataprocessing. The results presented in Example 1 (FIGS. 3 and 4)demonstrate the ability of the predictive-kinetic method of the presentinvention to accurately estimate end-point responses using aglucose-specific device, e.g., the biographer. End-point analyte-relatedvalues may then be correlated to glucose concentration or amount.

[0185] The results presented in Example 2 show that data using threehalf-lives provide a reliable estimate of the end-point charge. Further,a variance ratio was used to verify that the first-order model used wasa valid option. Similar data treatment demonstrated that using adiffusion-limited, flux model (e.g., Olsson, B., et al., Anal. Chem.58:1046-1052, 1986) also provided a valid predictive-kinetic model forthe data. For the benchtop studies performed with the biographer thefirst-order appears to provide the best estimates of the equilibriumvalues. However, other models may be applied to evaluation of the dataas described herein, such models include, but are not limited todiffusion limited models, a parallel multiple first order and an n-thorder model that does not require knowledge of the reaction order. Theparallel multiple first order can be expressed as follows:

S _(t) =S _(o)+(S _(∞) −S _(o))*(1 31 e ^(−k) ^(₁) ^(t))+(S _(∞2) −S_(o))*(1−e ^(−k) ^(₂) ^(t))+(S _(∞3) −S _(o))*(1−e ^(−k) ^(₃) ^(t))+ . ..  (Eqn. 6A)

[0186] where S_(o), S_(t) are initial and intermediate signals, S_(∞1),S_(∞2), S_(∞3), etc., are final (or end-point) signals (related to k₁,k₂, k₃, etc., respectively), k₁, k₂, k₃, etc., are the observedfirst-order rate constants, and t is time. This model is discussedfurther hereinbelow. This model is typically used in a situation wheremore than one first order reaction or process is occurring in parallel.In the model presented above in Eqn. 1, the optimized value of S_(∞)provides the predicted end-point value. As described herein, thepredicted end-point value may be employed in several ways. In oneaspect, the end-point analyte related value provides a measurementcorrelated to the amount or concentration of the analyte. Depending onthe application this value may be modified, for example, by addition orsubtraction of the initial signal and/or by applying calibration methods(which, for example, converts the value from current or charge toanalyte amount or concentration). Using the model presented in Eqn. 6A,the end-point signal is typically represented by the following equation:

S _(∞)=(S _(∞1) +S _(∞2) +S _(∞3)+ . . . )+S _(o)  (Eqn. 6B)

[0187] Whereas the final signal change in this case is typicallyrepresented by the following relationship:

ΔS _(∞)=(S _(∞1) +S _(∞2) +S ₂₈ ₃+ . . . )  (Eqn. 6C)

[0188] Based on the teachings of the present specification and knowledgeof one of ordinary skill in the art, a practitioner can choose whichembodiment of the end-point value (e.g., S_(∞) or ΔS_(∞)) betterrepresents the end-point value of interest. This choice may be guided,for example, by comparison of an end-point value to a calibration value,determined by independent methods, at a selected time point. Anotherapproach is to plot end point values (S_(∞) or ΔS_(∞)) vs. known analyteconcentration and utilize a determination of imprecision to select thebest end point value. In some situations, for example where there is ahigh initial signal (S_(o)) which is largely attributed to background,the ΔS_(∞) value may provide a better end-point value to use indetermination of the corresponding amount or concentration of analyte.

[0189] In some cases, for example, where background predominates theS_(o) value and when a background correction has been performed on thedata before application of the predictive-kinetic method of the presentinvention, the parallel multiple first order (Eqn. 6A) may be expressedas follows:

S _(t) S′ _(o)+(S _(∞1))*(1−e ^(k) ^(₁) ^(t))+(S _(∞2))*(1−e ^(k) ^(₂)^(t))+(S_(∞3))*(1−e ^(k) ^(₃) ^(t))+ . . .  (Eqn. 7A)

[0190] where S′_(∞), is an estimate of initial signal at t=0 (i.e.,S′_(o) corresponds to S_(o) after correction for the contribution ofbackground signal), S_(t) is an intermediate signal, S_(∞1), S_(∞2),S_(∞3), etc., are final (or end-point) signals (related to k₁, k₂, k₃,etc., respectively), k₁, k₂, k₃, etc., are the observed first-order rateconstants, and t is time. In this example, an estimate of S_(∞) and thecorresponding ΔS_(∞) may be represented as follows:

S _(∞)=(S_(∞1) +S _(∞2) +S _(∞3)+ . . . )  (Eqn. 7B)

ΔS _(∞)=(S _(∞1) +S _(∞2) +S _(∞3)+ . . . )−S′_(o)  (Eqn. 7C)

[0191] Typically in the methods of the present invention, the predictedend-point value is represented by S_(∞) whereas the change in thepredicted end-point value relative to the initial signal is representedas ΔS_(∞).

[0192] An n^(th) order model can be expressed as follows:

S _(t) =S _(∞)(−/+)[k(n−1)*t(+/−)(S _(∞) −S_(o))^(1−n)]^(1/(1−n))  (Eqn. 8)

[0193] where S_(o), S_(t), and S_(∞) are initial, intermediate, andfinal (or end-point) signals, k and t are the observed rate constant andtime, n is the order of the process, where n does not equal 1, and for(−/+) the first function (−) is used for data that increase in magnitudeas a function of time, and the second function (+) is used for thereverse case, correspondingly for (+/−) the first function (+) is usedfor data that increase in magnitude as a function of time, and thesecond function (−) is used for the reverse case.

[0194] An exemplary flux model can be expressed as follows:$\begin{matrix}{S_{t} = {S_{o} + {\left( {S_{\infty} - S_{o}} \right)\left\lbrack {1 + {2{\sum\limits_{i = 0}^{\infty}{\left( {- 1} \right)^{i}{\exp \left( {{- k_{1}}t} \right)}}}}} \right\rbrack}}} & \text{(Eqn.~~9)}\end{matrix}$

[0195] where S_(o), S_(t), and S_(∞) are initial, intermediate, andfinal (or end-point) signals, k₁=k_(i) ²π², k is the characteristicdiffusion rate constant, t is time, and i is a dummy-variable. In thismodel, the k₁ values should vary by π₂ (i.e., approximately 10). Thisvariation can be used as a control.

[0196] B. Reduced Dependency of the Predictive-Kinetic Model onMeasurement Variables

[0197] Temperature was selected as a variable to demonstrate the reduceddependency of the predictive-kinetic method on measurement variables.This variable was selected because it affects the rate of mutarotationas well as the rate of physical processes such as diffusion of glucosethrough a hydrogel (Kurnik R. T., et al., Journal of the ElectrochemicalSociety 145 (1998) 4119-4125). The results of these experiments areshown in Example 4. Clearly, the predictive-kinetic method gaveconsistent results regardless of the measurement temperature. Thepercent change in results observed with the predictive-kinetic methodrelative to temperature change was negligible. Further, regardless ofthe rate constant, similar end-point values were computed using thepredictive-kinetic method.

[0198] Following here is a proposed explanation for the reduceddependency of the predictive-kinetic method on measurement variables.The following should not be construed to be the only mechanisticexplanation possible and is provided only to possibly clarifyunderstanding of some aspects of the invention. Because rates ofreactions or processes, but not extent, can be affected by suchvariables as temperature, responses of analytical systems that are basedon kinetic measurements (e.g., electrochemical current generated by abiosensor) exhibit dependencies on these variables. Thepredictive-kinetic model as used to measure an equilibrium or end-pointcondition has much lower dependencies on these variables because therate of reaction controls the speed at which the system reaches theequilibrium point or end-point, but not the point itself

[0199] The predictive-kinetic model uses kinetic data to predict theequilibrium or end-point of a reaction. This model finds the best-fitvalue of several parameters, including the rate constant(s) of theprocess(es) involved in producing the signal. Because the rate constantis one of the fit parameters it provides a correction for variables thataffect the kinetic measurements, such as, temperature, diffusionconstants, enzyme kinetics, etc. For example, in the case of glucosemeasurement using an analyte monitoring device as described here, thedata set for each biosensor measurement cycle is fit individually.Accordingly, if the temperature changes occur during or between cyclesthen the model will fit a slightly different rate constant (e.g.,Example 4).

[0200] Sweat can contain certain analytes, such as glucose, and thuspotentially add a spurious signal to the analyte measurement and,accordingly, could give rise to inaccurate values. This problem may beavoided by, for example, incorporating a probe in the device to measureand indicate the presence of sweat on the skin (e.g., the GlucoWatchbiographer). However, because the predictive-kinetic method uses data inthe leading edge of the response cycle (e.g., the first threehalf-lives), sweat episodes that occur only after this period would notaffect the measurement and would allow display of that result to theuser. Evaluation of the half-life profile of data from several diabeticand non-diabetic subjects showed that three minutes of the transientresponse is sufficient to predict the end-point signals. Therefore,sweat episodes that occur after three minutes into the response cyclewill not affect the measurement.

[0201] Drift of values determined by an electrochemicalcurrent/biosensor system can also be significantly compensated by theequilibrium or end-point based measurement approach employed in thepredictive-kinetic method. This approach has reduced variabledependencies on such variables as temperature, hydrogel membranethickness, electrode kinetics, and enzyme activity such that changes inthese variables during or between measurement cycles do not affect thereliability and accuracy of the glucose measurements provided to the enduser.

[0202] A further example of the reduced dependency of thepredictive-kinetic method on measurement variables is illustrated by theability of the predictive-kinetic method to provide compensation fordeclining sensor signal. Factors responsible for decline of sensorsensitivity include, but are not limited to, the following: adsorptionof proteins on the electrode surface; and reduction in enzyme activity.Often, analyte monitoring devices that rely on platinum/carbon (Pt/C)electrodes and are worn by a subject for an extended period of time showa decline in sensor response over time. One possible ~explanation isattenuation in the signal caused by an apparent decline in sensitivityof the underlying Pt/C electrode.

[0203] Several approaches might be used to compensate for the decline insignal including, but not limited to, the following: (i) increasing themeasurement time for a given response cycle; and (ii) changing thehydrogel/sensor component after a predetermined time period of use.However, these approaches are typically neither cost effective norconvenient to the user.

[0204] The use of the predictive-kinetic method can compensate for theeffect of such sensor-based or enzyme-based signal decline in an analytemonitor because the method estimates the end-point signal consistentwith complete consumption of an extracted analyte, for example, glucosein a hydrogel. Accordingly, any decline in the sensitivity of the sensor(i.e., electrode) would not influence the predicted signal. Anillustration of the ability of the predictive-kinetic method tocompensate for such sensor-based signal decline is presented in Example5 with corresponding data in FIGS. 11 and 12. The results demonstratethat the predictive-kinetic approach can measure essentially all glucoseextracted, regardless of the sensor sensitivity, thereby compensatingfor the signal decline seen with a fixed integral measurement method.

[0205] These results demonstrate the usefulness of thepredictive-kinetic method applied to data obtained from a device thatprovides frequent and automatic analyte-related measurements. Theresults demonstrate that transient response from a such a device can bemodeled successfully and provide (i) a reliable estimate of asteady-state signal, (ii) calibration curves similar to a steady-statemodel, (iii) lower dependence on measurement variables.

[0206] When the biographer is being used by a subject there is anextraction period followed by a measurement period. During theextraction period most of the analyte (i.e., extracted glucose or itsconversion product, hydrogen peroxide) localizes near the reactive faceof the electrode. However, some of the analyte may be dispersedthroughout the hydrogel. When voltage is applied to the reactive face ofthe electrode in order to quantitate the signal from the analyte, thisresults in an initial current with a signal that decays following apseudo-first order rate constant (k₁); however, in this situation aparallel multiple first order reaction is taking place as the remainderof the analyte reaches the reactive face and generates current (rateconstant k₂). Accordingly, the parallel multiple first order modeldiscussed herein above provides one model for the biographer when it isbeing used in operative contact with a subject. In this embodiment, theparallel multiple first order can be expressed as follows:

S _(t) =S _(o)+(i S_(∞1) −S _(o))*(1−e ^(−k) ^(₁) ^(t))+(S _(∞2) −S_(o))*(1−e ^(−k) ^(₂) ^(t))  (Eqn. 10)

[0207] where S_(o), S_(t), S_(∞1), and S_(∞2) are initial, intermediate,and final signals (related to k₁ and k₂, respectively), k₁, k₂, and tare the observed first-order rate constants and time, respectively. Inthis case, the two rate processes that determine responses from thebiographer have significantly different magnitudes, k₁>>k₂. Typically,the ratio of k₁ to k₂ remains constant. Maintenance of this constantrelationship may be used as a criterion of good fit of the data.

[0208] Accordingly, when predicting the end-point value (e.g.,S_(∞1)=S_(o)+S_(∞1)+S_(∞2)) there are now a total of five parameters tobe estimated for each predicted S_(t), those parameters being S_(∞1),S_(o), k₁, S_(∞2), and k₂. As discussed above these parameters areestimated and predicted values of S_(t) iteratively generated until theerror between the predicted values and the actual data points isminimized. That is, the iterative process is repeated until errorminimization is achieved. The result is a final value for S_(∞) which isthen converted to glucose amount or concentration by, for example,multiplying the value with a calibration value.

[0209] In one embodiment, a calibration value can be determinedessentially as follows. S_(∞) is based on a measurement cycle of thebiographer at a calibration point, wherein in the corresponding timeframe the subject also performs, for example, a finger stick todetermine the blood glucose value at the calibration point. The amountof glucose at the calibration point can be determined using, forexample, a HemoCue® (Aktiebolaget Leo, Helsingborg, Sweden) clinicalanalyzer. The blood glucose measurement obtained is used as a singlepoint calibration, which is used to calculate the extracted bloodglucose amounts or concentrations for all subsequent GlucoWatchbiographer measurements. Accordingly, the calibration value is equal tothe measured blood glucose amount at the calibration time divided by thepredicted nC value determined at the calibration time (i.e., S_(∞)).Subsequent predicted nC values (i.e., S_(∞)) are then multiplied by thiscalibration value to obtain blood glucose amount or concentration.

[0210] C. Optimization of Signal Measurement Time

[0211] In addition to the predictive-kinetic method described herein,the present invention also includes methods to determine if enough datapoints have been gathered by the biographer in order to produce reliablepredicted values. Experiments performed in support of the presentinvention suggest that data in the kinetic portion of the curvecorresponding to three or more half lives of the signal provide reliablepredicted values. Accordingly, the time period through which thebiographer is measuring signal can be dynamically evaluated while signalis being measured and collected. For example, typically at least threedata points are collected. These data points are used to estimate afirst order rate constant k by plotting the natural log of signal(S_(t)−S_(o)) over time, where the slope of the resulting linecorresponds to an estimate of k. To simplify, this relationship can beexpressed by the following equation: t_(½)=ln 2/k (i.e.,t_(½)=0.6293/k).

[0212] Based on empirical observations an average optimal measurementtime can typically be determined (e.g., three minutes for thebiographer). However, in an alternate embodiment to ensure that at leastthree half lives of the signal are encompassed by this time period, analgorithm can be established that calculates t_(½) for the signal data.This value is then multiplied by three. If the resulting value is lessthan the average optimal measurement time, then that measurement time issufficient. If, however, the resulting value is greater than the averageoptimal measurement time, then the biographer can be instructed by thealgorithm to continue its measurement cycle until three half lives ofthe signal (or a finite cut-off point) is achieved.

[0213] Accordingly, in a preferred embodiment of the present inventionthe steps of a method for determining blood glucose concentration oramount are as follows:

[0214] (i) collect current data (e.g., at least three signal values inthe kinetic range);

[0215] (ii) estimate the rate constant (k) for a first order model byplotting the natural log of signal (S_(t)−S_(o)) over time, where theslope of the resulting line corresponds to an estimate of k;

[0216] (iii) estimate the half-life of the signal using t_(½)=ln 2/k;

[0217] (iv) (a) if three times the resulting value is less than theaverage optimal measurement time, then that measurement time issufficient. (b) If three times the resulting half-life value is greaterthan the average optimal measurement time, then the biographer isinstructed by an algorithm to continue its measurement cycle until threehalf lives of the signal (or a finite cut-off point) is achieved;

[0218] (v) the current data is integrated (resulting in nCdata)—appropriate background subtraction may be performed before thisstep if desired;

[0219] (vi) the actual nC data is used to estimate parameters in theselected model, for example, the parallel multiple first order model,and an error is determined (e.g., sum of squares for predicted valuesminus actual values). This process is repeated (i.e., an iterativeprocess) until either the error is less than a predetermined thresholderror (determined, for example, using change in chi-square less than0.05%) or there is no further change in the error upon furtheriteration. This error minimization steps can be carried out by a numberof methods known in the art, for example, the Levenberg-Marquardt Methodor simplex optimization method. (See, for example, error minimizationmethods described in Numerical Recipes in C, Second Edition, CambridgeUniv. Press, 1992.)

[0220] (vii) the final predicted S_(∞) is then converted to a bloodglucose value by multiplying the predicted S_(∞) by a calibration value.

[0221] This method can be adapted by one of ordinary skill in the art,following the guidance of the specification in combination with what isknown in the art, to employ different models that are selected to bestrepresent the signal for a selected analyte/measurement system (forexample, by changing the model in step (vi)).

[0222] D. Variations on the Parallel Multiple First Order Model

[0223] The parallel multiple first order model (PMFOM) assumes more thanone first order process occurs simultaneously. The PMFOM finds thebest-fit of the data by deconvoluting the measured data into separateprocesses, i.e., there is a segregation of processes. The best-fitparameters are then used to predict the end-point nC value of theintegrated signal. However, certain components of the signal may notarise from glucose, but may be due to other electrochemical processes,e.g., Pt oxidation, background currents from impurities, interferingspecies, etc. In the context of determining glucose analyte amounts orconcentration, only the signal (i.e., charge) arising from the glucoseis of interest; other components contributing to charge are,essentially, noise.

[0224] With a sufficiently high density of current points during themeasurement cycle, with which to fit the data, it is possible to includein the PMFOM as many parallel processes as are identified. In this way,the total signal is separated into its individual components, i.e., eachcomponent arising from a different process. Then only those componentsshown to arise from glucose can be used in the PMFOM.

[0225] For example, suppose that a PMFOM with three parallel processesis used. The total signal at time t is then the sum of the threeindividual processes:

S _(t) −S _(o)+(S _(∞1) −S _(o))*(1−e ^(k1t))+(S _(∞2) −S _(o))*(1e^(−k2t))+(S _(∞3) S _(o))*(1−e ^(−k3t))  (Eqn. 11)

[0226] where S_(o), S_(t) are initial and intermediate signals, S_(∞1),S_(∞2), S_(∞3), are final (or end-point) signals (related to k₁, k₂, k₃,respectively), k₁, k₂, k₃, are the observed first-order rate constants,and t is time.

[0227] As described above, in some cases, for example, where backgroundpredominates the S_(o) value and when a background correction has beenperformed on the data before application of the predictive-kineticmethod of the present invention, the parallel multiple first order maybe expressed as follows:

S_(t) =S′ _(o)+(S _(∞1))*(1−e ^(−k) ^(₁) ^(t))+(S _(∞2))*(1−e ^(k) ^(₂)^(t))+(S _(∞3))*(1−e ^(−k) ^(₃) ^(t))  (Eqn. 12)

[0228] where S′_(o), is an estimate of initial signal at t=0 (i.e.,S′_(o) corresponds to S_(o) after correction for the contribution ofbackground signal), S_(t) is an intermediate signal, S_(∞1), S_(∞2),S_(∞3) are final (or end-point) signals (related to k₁, k₂, k₃respectively), k₁, k₂, k₃, are the observed first-order rate constants,and t is time. In this example, an estimate of S_(∞) may be representedas follows:

S _(∞) =S _(∞1) +S _(∞2) +S _(∞3)  (Eqn. 13A)

ΔS _(∞=() S ₁ +S _(∞2) +S _(∞3))−S′_(o)  (Eqn. 13B)

[0229] The relationships shown in Eqn. 13A and Eqn. 13B are applicableto one embodiment of the invention where a “previous”baseline-subtracted current is being used in the modeling. For example,in the case of the GlucoWatch biographer there is a previous cycle “A”during which the biosensor is at the iontophoretic anode, and a presentmeasurement cycle “B” during which the biosensor is at the iontophoreticcathode. The last points (e.g., two points) from the previous cycle “A”are used as a measure of the baseline background current. This baselinevalue is then subtracted from all the current values obtained in thepresent measurement cycle “B” before integration of those values. Atemperature correction of the previous baseline value may be performedprior to the subtraction step.

[0230] Further, empirically it may be determined, for example, that oneof the three processes (e.g., (S_(∞1))*(1−e^(k1t))) has littlecorrelation with blood glucose amount or concentration. The method thencould segregate such a process from the rest of the terms. Accordingly,in this situation, a better fit to the blood glucose data may berepresented by the following relationship:

S _(t) =S′ _(o)+(S _(∞2))*(1−e ^(−k2t))+(S _(∞3))*(1−e ^(−k3t))  (Eqn.14A)

[0231] In this situation (Eqn. 14) the end-point value would berepresented as follows:

S _(∞) =S _(∞2) +S _(∞3)  (Eqn. 14B)

ΔS _(∞)=(S _(∞2) +S _(∞3))−S′_(o)  (Eqn. 14C)

[0232] In another example, empirically it may be determined a better fitmay be obtained while including S_(o) and eliminating the contributionof one of the processes, that is, a process (e.g.,(S_(∞1))*(1−e^(−k1t))) has little correlation with blood glucose amountor concentration, for example:

S _(t) =S _(o)+(S _(∞2) S _(o))*(1−e ^(−k2t))+(S _(∞3) −S _(o))*(1−e^(−k3t))  (Eqn. 15A)

[0233] where S_(o), S_(t) are initial and intermediate signals,respectively, S_(∞2), S_(∞3), are final (or end-point) signals (relatedto k₂, k₃, respectively), k₂, k₃, are the observed first-order rateconstants, and t is time. In this situation (Eqn. 15A) the end-pointvalue would be represented as follows:

S _(∞)−(S _(∞2) +S _(∞3))+S_(o)  (Eqn. 15B)

ΔS _(∞)(S _(∞2) +S _(∞3))  (Eqn. 15C)

[0234] Accordingly, the relationship between each process {e.g.,(S_(∞1)−S_(o))*(1−e^(−k1t)), (S_(∞2)−S_(o))*(1−e^(−k2t)), and(S_(∞3)−S_(o))*(1−e^(−k3t))} and an analyte value can be determinedstatistically by examining the contribution of each process to the totalsignal, and its correlation to the analyte amount or concentration, forexample, the blood glucose measurement. The value of the contribution ofthat process to the overall determination of the blood glucosemeasurement may be decided. Alternatively, if the process can beidentified with a known process (e.g., mutarotation, Pt oxidation, etc.)the correlation can be determined from first principles.

[0235] However, in view of the above, measured current data (i.e., notbaseline-subtracted) may be used as the input for the model. In thissituation, the background correction would be accomplished by combiningthe background into the S_(o) term. Alternatively, the background may befit into a first or zero order type of behavior if the transient portionof the background is taken into account. Using the measured current data(i.e., not baseline-subtracted) eliminates error due to improperbackground subtraction arising, for example, from skin permeabilitydifferences, incomplete consumption of glucose, as well as,interferences in the anode baseline, sensor noise, or differentsensitivities between two sensors in a two sensor system. In a furtherembodiment, the background may be included as a term in the predictivekinetics, where it is not limited to a first order model, e.g., it couldbe a zero-order or quadratic-order term. An example of a model includinga zero-order term is as follows:

S _(t) =S _(o) +k _(o) t+(S _(∞1) −S _(o))*(1−e ^(−k1t))+(S _(∞2) −S_(o))*(1−e ^(−k2t))+(S _(∞3)S_(o))*(1−e ^(−k3t))  (Eqn. 16)

[0236] where k_(o) is a pseudo-zero order rate constant, and the otherterms are as described above (e.g., Eqn. 12). Such an approach may beused to resolve the experimental responses into different components,where, for example, one component represents a zero order term.

[0237] In yet another embodiment of the present invention, a weightingfactor (ω_(x)) may be used to give different weights to the differentprocesses to improve correlation with blood glucose value. Suchweighting factors skew the contribution of the corresponding process toaccount for the level of contribution of each process to the overalldetermination of blood glucose value, that is, weighting factors reflectthe relative importance of the process with regard to the overalldetermination. The sum of the weighting factors is typically equal toone (i.e., Σ(ω_(x))=1, where X is the number of processes). For example,a weighted, three process determination may be represented as follows:

S _(t)=ω_(o) S _(o)+ω₁(S _(∞1) −S _(o))*(1−e ^(−k1t))+ω₂(S _(∞2) −S_(o))*(1−e ^(−k2t))+ω₃(S _(∞3) −S _(o))*(1−e ^(−k3t))  (Eqn. 17)

[0238] where ω₀, ω₁, ω₂, and ω₃ are weighting factors andΣ(ω₀+ω₁+ω₂+ω₃)=1.

[0239] Another approach to baseline correction is to fit a suitablemodel to the measured current data curve (e.g., i vs. t, where i iscurrent and t is time) and use the predicted, end-point, baseline valueto perform background subtraction. In the case of the Gluco Watchbiographer (as an example of a two sensor system), iontophoreticextraction takes place into an anodic and a cathodic reservoir, each ofwhich is in contact with a sensor element. The majority of sampledglucose is located in the cathodic reservoir. An anodic detection cycleis performed which results in a response curve comprising data pointsfrom which, using the predictive-kinetic methods of the presentinvention, a baseline, end-point, background value can be predicted. Thecathodic detection cycle is then performed. The predicted end-pointbackground value from the anodic cycle may then be used for backgroundsubtraction of the data obtained in the cathodic detection cycle (e.g.,the predicted end-point background value may be subtracted from eachdata point of the cathodic detection cycle response curve in order toprovide a background corrected response curve for the cathodic detectioncycle). This approach compensates for, e.g., incomplete reactions in theanodic half-cycle during operation of the Gluco Watch biographer.

[0240] In the current GlucoWatch biographer, a background correction ofthe cathodic cycle values is performed using an average of at least thelast two data points from the current response curve obtained in theanodic cycle measurement which preceded the cathodic cycle. Thisbackground correction is performed before integration of the cathodicresponse (i.e., the background corrected cathodic response data isintegrated). In some cases, a negative deviation (FIG. 20A) of thecorrected cathodic response is observed when the anodic current valueused for the background correction is greater than the cathodicresponses near the end of the measurement period (which is typicallyseven minutes). Conversely, a positive deviation (FIG. 20B) is observedwhen the anodic current is lower than the cathodic responses.

[0241] A final background value (i.e., the anodic or cathodic responsesat an infinite time point) can be estimated using a suitable model(e.g., first order, second order, parallel first order, etc.) applied inthe predictive-kinetic method of the present invention. This valueprovides compensation for an incomplete reaction or unstable baselinemeasured during the anodic or cathodic cycles. Use of this method ofbackground subtraction may provide an integrated response for thecathodic cycle that requires no further treatment. However, applicationof the predictive-kinetic method of the present invention to theintegrated cathodic cycle response curve to predict an end-pointanalyte-related value may still make a meaningful contribution toaccurate prediction of the end-point analyte-related value.

[0242] Depending on the data processing method utilized, incompletereaction of analyte, or an unstable signal, may affect the analyticalperformance of a device performing frequent measurements, such as theGluco Watch biographer glucose monitor. For instance, performance isaffected when a signal value inconsistent with complete reaction is usedto perform background correction. This causes attenuation during thelatter period of the response curve, especially for low analyteconcentration. One approach to compensate for this is to allow thereaction to go to completion. However, this is may require longmeasurement time, which is incompatible with providing timelyinformation from a monitoring device, for example, for providinginformation regarding glucose excursions.

[0243] As discussed above, an alternate method is to estimate a signalvalue consistent with the signal value of the complete reaction byfitting a suitable model to the response. The estimated value is anaccurate measure of signal at the completion of reaction and isindependent of analyte concentration. This provides a reliable estimateof analyte concentration achieved by subtracting the predicted finalbackground signal from the values of the fitted line of the responsecurve prior to integration to estimate the area under the responsecurve. Use of the fitted line rather than raw data provides additionalbenefit of signal averaging that reduces noise beyond the levelattainable by simply integrating the corrected current response. Thisapproach also reduces the influence of systematic noise, such astemperature spikes.

[0244] This data processing option was evaluated by fitting a suitablemodel to the current versus time response of the biographer glucosemonitor (Example 6). The model selected was an exponentially decayingsignal consisting of two rate processes as shown below:

S _(t) =S _(o) +S ₁ *e ^(−k1*t) +S ₂ *e ^(−k2*t)+final_Bkgrd  Eqn. 20

[0245] where S₀ is response at t=0, S₁ and S₂ were the signalsconsistent with the two processes with pseudo (or apparent) rateconstants were k₁ and k₂, and t was time. Final_bkgrd was the estimatedsignal response at completion of the reaction. In some cases, such as inthe case where there is a large transient background current, So may beignored and Eqn. 20 then becomes:

S _(t) =S ₁ *e ^(−k1t) +S ₂ *e ^(−k2*t)+final_Bkgrd  Eqn. 21

[0246] Experiments employing Eqn. 21 to estimate end-point backgroundvalues and resulting corrected data curves are presented and discussedin Example 6. Fits of data (using the predictive-kinetic method of thepresent invention, for example employing Eqn. 21) to the current versustime response and subsequent data treatment as described herein allowsfor a reliable estimate of equilibrium value consistent with completeconsumption of analyte, for example, glucose. Because this methodestimates total analyte consumed, it provides an invaluable tool toexamine decline in sensitivity of the response of a monitor to analyteover an extended period.

[0247] E. Specialized Algorithms

[0248] In yet another aspect of the present invention, prediction of theconcentration of an analyte can be accomplished using specializedalgorithms, where the specialized algorithms are useful for predictionsin particular situations (e.g., particular data sets or ranges ofpredicted values) and where the algorithm used for performing thecalculations is determined based on the situation. In this case a“switch” can be used to employ one algorithm (or group of algorithms)rather than another algorithm (or group of algorithms). For example, inone embodiment of the present invention an algorithm is used todetermine if enough data has been collected to obtain accuratemeasurements using the predictive-kinetic method of the presentinvention. First, current data is collected (e.g., at least three signalvalues in the kinetic range). A rate constant (k) is estimated for afirst order model by plotting the natural log of signal (S_(t)S_(o))versus time, where the slope of the resulting line corresponds to anestimate of k. The half-life of the process is then estimated using, forexample, t_(½)=ln 2/k. A “switch” in the algorithm is used as follows:(a) if three times the resulting value for the half-life is less thanwhat has empirically been determined to be the average optimalmeasurement time, then that measurement time is considered to besufficient and the data is employed by the predictive-kinetic method toestimate an end-point value. However, if (b) the three times resultingvalue for the half-life is greater than the average optimal measurementtime, then the biographer is instructed by an algorithm to continue itsmeasurement cycle until three half lives of the signal (or a finitecut-off point) is achieved, before proceeding to the predictive-kineticmethod to estimate an end-point value.

[0249] Initial estimates of parameters that vary significantly from thereal values (i.e., measured values) can cause problems ofnon-convergence and in some cases require many iterations to achieveconvergence. One solution to this problem is to use empirical methods tocompute initial estimates, such as, linear regression to estimate a rateconstant from a plot of In (S_(t)−S₀) vs t. This method is useful whenthe range of the data being used generally follows first order response;otherwise inaccurate estimate of the rate constant can result.

[0250] For cases where the responses are non-first order, a differentapproach, such as Guggenheim method (Guggenheim, E. A. Philos. Mag. J.Sci.; 1926, 2, 538) may be employed. The Guggenheim method assumes noknowledge of the response profile and uses an algorithm to estimate therate constant. In the Guggenheim method, several pairs of responses,S_(i) and S_(j) are measured, with each pair being separated by the samefixed time interval, t. Then In (S_(i)−S_(j)) vs t₁ is plotted to obtaina linear plot with intercept on the Y-axis equal to In S_(o), from whichS_(o) is computed. The form of the equation used in the process is asfollows:

ln(S ₁ −S _(n))=−kt ₁+ln[(S _(o) −S _(SS))(1−e ^(−kΔt))]  (Eqn. 18)

[0251] where S₁ is the response measured at the first point in time, t₁,and S_(n) is the response measured at the last point in time, t_(n), kis the slope of the line, S_(SS) is the steady-state response.

[0252] The method of partial sums (see, e.g., Cornell, R. G. Biometrics(1962) 18:104-113) may also be employed to provide reliable values ofinitial estimates of the rate constant without prior knowledge ofresponse at t=0 (S_(o)) or time t (S_(t)).

[0253] In most cases, either the Guggenheim or partial sum methodproduces only one distinct value for the rate constant, then a secondrate constant, for example for a two process parallel first order model,may be obtained using the first rate constant. One approach toaccomplish this is to multiply the first rate constant value by aselected factor.

[0254] Once accurate initial estimates of the rate constants areobtained, then initial values of other parameters may then be computedby multiple regression with the equation below for a parallel firstorder model, for example, as follows:

S _(t) =S _(o) +S _(∞1)[1−exp(−k₁ °t)]+S∞, ₂[1−exp(−k₂ °t)]  (Eqn. 19)

[0255] where k₁° and k₂° are the computed initial estimates of rateconstants for a parallel first order model. Other parameters in thisequation have been described above. Finally these initial estimates areused with, for example, Eqn. 6A to predict best-fit values of the finalresponses.

[0256] Further, different models employed in the predictive-kineticmethod of the present invention may be used under differentcircumstances. In this case, a more global algorithm can be the switchused to selected one of several different algorithms (e.g., switchingbetween a first order model and a parallel multiple first order model).In one embodiment such a global algorithm may be used to determine apreliminary blood glucose value. The blood glucose value is determined,by the algorithm, to fall into one of three ranges (for example, low,normal, and high). For each range there is an separatepredictive-kinetic algorithm that optimizes the prediction for values inthe particular range.

[0257] Specialized algorithms may be developed to be used in differentparts of a range of analyte signal spectrum or other input values (e.g.,for all parameters used in the prediction). A global algorithm can beused to decide which region of the spectrum the analyte signal is in,and then the global algorithm switches the data to the appropriatespecialized algorithm. Further, there can be multiple levels ofspecialized switching (which can be graphically represented, forinstance, by branched tree diagrams).

[0258] Experimental

[0259] The following examples are put forth so as to provide those ofordinary skill in the art with a complete disclosure and description ofhow to make and use the devices, methods, and formulae of the presentinvention, and are not intended to limit the scope of what the inventorregards as the invention. Efforts have been made to ensure accuracy withrespect to numbers used (e.g., amounts, temperature, etc.) but someexperimental errors and deviations should be accounted for. Unlessindicated otherwise, parts are parts by weight, molecular weight isweight average molecular weight, temperature is in degrees Centigrade,and pressure is at or near atmospheric.

EXAMPLE 1 Preliminary Studies

[0260] The data was collected at room temperature using the GlucoWatchbiographer. In these studies, the biosensor was assembled andpreconditioned for one hour. Ten microliters of glucose solution ofknown concentration was then deposited on the hydrogel. The electroderesponse to the glucose was monitored for 60 minutes. The currentresponses were then transferred to a computer for data processing.

[0261] In FIGS. 3 and 4, the points are experimental data for responseof the biographer to 200 μmol/L glucose and the solid curve is fit tothe data using the first-order model:

S _(t) =S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1)

[0262] where S_(o), S_(t), and S_(∞) are initial, intermediate, andfinal signals, k and t are the first-order rate constant and time,respectively. The experimental response was obtained by first applyingpreconditioning potentials at 0.77V vs. Ag/AgCl for 10 min, followed bya step down to 0.42V vs. Ag/AgCl for 50 min. At the end of the 60 minpreconditioning period, 10 μL of the 200 μmol/L glucose solution wasdeposited on the hydrogel and the response was monitored to completion.The current measured after the solution was deposited was integrated andis shown in FIGS. 3 and 4 as charge vs. time response.

[0263] The first-order model was fit only to the data during the firstthree half-lives (open circles) of the process. By extrapolating the fitbackward to t=0 and forward to t≧10t_(½) it is possible to obtainpredicted values of the initial and final values of the signal S_(o) andS_(∞), respectively. Agreement between computed and measured results isillustrated in FIG. 4 which contains more experimental data points nearequilibrium. By using the predicted values, it is possible to computethe signal change, ΔS=S_(∞)−S_(o), that would have been measured had theprocess been monitored from t=0 to equilibrium or completion. Thispredicted change varies linearly with glucose concentration.

[0264] Further, charge versus time responses for different glucoseconcentrations were also evaluated. Glucose solutions of differentconcentrations were evaluated as described above. The data are presentedin FIG. 5. In the figure, dots represent data points and the lines thefitted curves using Eqn. 1 and an error minimization protocol. Thefitted lines were calculated using three half-lives of the signal. Theresults demonstrate the high correlation of the predictive-kineticmethod of the present invention to glucose concentration or amount inthe sample as detected by the biographer.

[0265] These results demonstrate the ability of the predictive-kineticmethod of the present invention to accurately estimate glucoseconcentration or amount based on the data provided by the biographer.

EXAMPLE 2 Further Modeling Studies

[0266] In this study the responses of the biographer to differentconcentrations of glucose (25 to 200 μmol/L) was monitored using theprocedure described in Example 1. The concentration range covered theextracted glucose values, as determined using a biographer, found inpatients with diabetes. The experiment was performed at room temperatureusing sensors of moderate sensitivity. There were six replicatemeasurements at each concentration. The first-order model was fitted tothe integrated data using several half-lives. The first half-life was250 secs. FIG. 6 presents a plot of the predicted charge, obtained withthe predictive-kinetic method, vs. the theoretical charge.

[0267] During the measurement of glucose, the biosensor current isintegrated as a function of time. The integral of an electrical currentis an electrical charge (Q=I×t). Because the total amount of glucoseadded to the biosensor in this experiment and the number of electronsreleased during the electrochemical reaction (2 per glucose molecule)were known, the theoretical charge was calculated by the Faradayequation:

Q=N×n×F×10

[0268] where N=concentration of glucose, μmol/L deposited on thehydrogel; n=number of electrons released per mole of glucose; F=96,500C/mole, i.e., Faraday's constant; and 10 equals the volume of glucose,μl.

[0269]FIG. 6 shows a plot of predicted versus theoretical charge and theslope value which represents the extent of recovery of the glucoseconcentration deposited on the hydrogel was >93% (this value wasobtained from the slope of a plot of predicted charge vs. theoreticalcharge shown in FIG. 6, where the slope was approximately 0.929 orapproximately 93%). When the theoretical charge was achieved, it wastermed 100% recovery of the analyte, and was considered the end-point ofthe reaction. Theoretical charge is plotted on the x-axis in FIG. 6. Theend-point charge predicted by the predictive-kinetic model using threehalf-lives of the response data, are plotted on the y-axis.

[0270] For the 200 μmol/L sample, average of charge estimated by thepredictive-kinetic method was 363,930 nC compared to an expected valueof 386,000 nC, >or 94% which is close to the average of 93% determinedfrom the slope of the line in FIG. 6. This shows that thepredictive-kinetic method can, on average, estimate 93% of themeasurement objective consistent with complete glucose consumption. Thepredictive-kinetic method also gave results with low imprecision(S_(y.x)=2491 nC ).

[0271] Results for other fitting ranges are shown in Table 1. Thesevalues were obtained from a plot of predicted charge versusconcentration (micromole per liter). The slope value is given innC/μmole/L. TABLE 1 Slope Pooled Vari- (nC/ Intercept Sy.x S.D anceMethod μmol/L) (nC) R² (nC) (nC) Ratio First-Order: 1^(st) half life2315 −19697  0.9442 31167  32620  0.91 2^(nd) half life 1829 −43130.9975 5242 5355 0.96 3^(rd) half life 1785 −3131 0.9993 2768 2894 0.914^(th) half life 1777 −2968 0.9996 2064 2193 0.89 8^(th) half life 1791−5127 0.9996 2169 2233 0.94

[0272] The values in the slope column show that data using threehalf-lives provide a reliable estimate of the end-point charge. Alsoincluded are the pooled standard deviation and the variance ratiosestimated from the equation given below.

Variance ratio=S ² _(y.x) /SD ² _(pooled)

[0273] The variance ratio was used to verify that the first-order modelused was a valid option for this data set. The F_(table) at 95%Confidence Interval is 2.78 and since the variance ratios are all lessthan this value, it confirmed that the fit of the first-order model tothe data was valid.

[0274] Although the first order model data fits the empirical data well,other empirical models are available (including a parallel multiplefirst order and an n-th order model) which do not require knowledge ofthe reaction order and which can provide accurate predictions. Forexample, analyses of these data using alternative models (e.g., acombined zero order and first order model, as well as, adiffusion-limited flux model) also provided good fit of the data to themodels.

EXAMPLE 3 First Order Versus Parallel-First Order for Fitting ClinicalData

[0275] Modeling of clinical data obtained from patients with diabetesdemonstrated that the parallel multiple first order model fits clinicalsignal from the biographer's biosensor more accurately than first ordermodel. This is, at least in part, because the biographer is used in sucha fashion that involves at least two rate processes—initial reaction atthe electrode and at least one parallel reaction, in this case where thesecond parallel reaction may be mutarotation and diffusion dependent.During the three-minute extraction cycle used by the biographer, glucoseaccumulates in the hydrogel near the reactive face of the biosensor, andonce the potential is applied a large current response is observed. Thisgives rise to the first rate process. Further rate processes can be dueto other factors such as mutarotation and diffusion of glucose throughthe hydrogel. These further processes are typically much slowerprocesses. In the present case, the second rate process consideredprimarily resulted from diffusion.

[0276] The data was collected at room temperature using the GlucoWatchbiographer worn by a non-diabetic test subject. The data presented inFIGS. 7 and 8 show typical fits of models to signals (i.e., a singlecharge measurement predicted over a three minute time period presentedin nC) obtained using the biographer. The fits are of a first ordermodel (Eqn. 1 above; to data shown in FIG. 7) and a parallel multiplefirst order model (Eqn. 10 above; to data shown in FIG. 8). Theimprovement in the fit using the parallel-first order model is shown bylower χ² value (165) and higher value (29197) for the First Order model(see legends in FIGS. 8 and 7, respectively). Further, the ratio of k1to k2 may be used to determine the quality of the fit. For example, inthe present analysis, based on bench-top data the ratio was expected tobe about 9±2. If the fit of a model (Eqn. 21) to a response gives aratio significantly different from 9±2 it makes the quality of thatparticular response reading questionable and, accordingly, may beeliminated as an erroneous reading.

[0277] After 3 hours of data were collected, the subject took a largedose of oral glucose. The data from the experiment is presented in FIG.9. In the figure, triangles show the values obtained using a standardfinger prick method and conventional meter to determine blood glucose,the scale for this measurement is the Blood Glucose (secondary Y-axis);squares show data gathered using the biographer employing a fixed pointmeasurement method (the fixed point determination being made after 7minutes of signal measurement), the scale for this measurement ispresented on the Charge axis; and diamonds show data gathered using thebiographer employing a three minute measurement and thepredictive-kinetic method of the present invention using a parallelmultiple first order model (Eqn. 10, above), the scale for thismeasurement is presented on the Charge axis. The plots in FIG. 9 presentresults for all extraction/measurement cycles during an eight-hour test.

[0278] The signals predicted by the parallel-first order and the fixedpoint methods both tracked with the meter-estimated blood; however, thesignals predicted by the parallel multiple first order used only threeminutes of the collected data for the prediction, as compared to 7minutes used for the fixed time method.

[0279] Finally, a plot of chi-square versus elapsed time for all cyclesclearly showed that both first order and parallel multiple first ordercan fit the time-dependent responses from a test subject for normalglucose levels at lower levels of blood glucose (FIG. 10). In thefigure, the data is broken down into measurements at the first andsecond sensors (i.e., sensors A and B) employed by the biographer. Usingthe biographer, a complete measurement cycle is typically as follows.Analyte is extracted from the test subject into a first hydrogel using athree minute iontophoretic extraction, followed by the sensing (ormeasurement) period for that hydrogel, i.e., determination of currentassociated with the amount or concentration of analyte present in thehydrogel. This cycle is repeated employing a second hydrogel.Accordingly, a “complete” measurement cycle includes the signal datafrom both hydrogels.

[0280] However, at higher glucose levels, the first-order did notprovide as reliable estimates of the end point values as the parallelmultiple first order which gave more consistent values (FIG. 10, timeperiod from approximately three to five hours). The increase inchi-square values for fits with the first order coincided with ingestionof an oral glucose drink by the test subjects at about three hours intothe study. This increase in glucose levels affected the results of firstorder model while the parallel multiple first order model continued toprovide reliable estimates of the end point values as shown by low andconstant chi-square values throughout the study period.

EXAMPLE 4 Variable Dependency Study

[0281] Temperature was selected as a variable to demonstrate the reduceddependency of the predictive-kinetic method on measurement variables.This variable was selected because it affects the rate of mutarotationas well as the rate of physical processes such as diffusion of glucosethrough a hydrogel (Kurnik R. T., et al., Journal of the ElectrochemicalSociety 145 (1998) 4119-4125). Data were collected with the biographerat 21° C. and 32° C.

[0282] The equilibrium charges at each temperature were estimated usingthe predictive-kinetic method by fitting the first-order model to thecharge vs. time responses. The data range used was three half-lives. Thedata showed that the same equilibrium charge was predicted for bothtemperatures.

[0283] Table 2 below contains average results of the study. Fourreplicate measurements were made at each temperature. The effectivenessof the predictive-kinetic approach to deal with temperature variationwas demonstrated by fitting a first-order model to data collected at twomeasurement temperatures (21° C. and 32° C.) for 200 μmol/L glucose.Even though the pseudo-first order rate constant increased from 2.3 to4.4×10⁻¹ (see Table 2 below), the predicted end-point signal onlyincreased by 6%. By comparison, non-end point methods, such as,integrated signals at a fixed-time of 10 minutes, showed an increase of26% in the response to the change 11° C. Clearly, the predictive-kineticmethod gave consistent results regardless of the measurementtemperature. The percent change observed with the predictive-kineticmethod was negligible. Another important observation was that,regardless of the rate constant, similar equilibrium values werecomputed using the predictive-kinetic method. This is shown by thepercent recoveries of 95% and 100% estimated at the two temperaturesusing the predictive-kinetic method. TABLE 2 21° C. 32° C. Ave (nC)367700 391435 Std. Dev 17196 5967 % CV 4.8 1.5 % Recovery 95 101 %Change/° C. — 0.6 Ave Rate Constant 2.3 4.4 (10⁻³, sec⁻¹)

EXAMPLE 5 Compensation of Declining Signal by the Predictive-KineticMethod

[0284] Response curves were established for the last four hours of (i) aglucose monitor employing a Pt/C electrode and the predictive-kineticmethod, and (ii) a glucose monitor employing a Pt/C electrode and afixed-time concentration determination method (fixed integralmeasurement). The predictive-kinetic method was based on a parallelmultiple first order response (see above, Eqn. 6A) using three minutesof signal measurement per data point. The fixed time determination wasbased on seven minutes of signal measurement per data point. Theresponse curves are presented in FIG. 11 Also in FIG. 11 data for bloodglucose amounts as determined using a OneTouch® (Johnson & Johnson, NewBrunswick, N.J.) device are presented in solid triangles with thereference axis being the right vertical axis.

[0285] The data presented in FIG. 11 clearly show an apparent decline inthe values obtained by the fixed-point method. The values predicted bythe kinetic model are higher for the same region. This difference indecay can be illustrated by plotting a ratio of the predicted values tothe fixed-point data. The ratios of the values in FIG. 11 are plotted asa function of elapsed time in the FIG. 12. Values greater than one wouldindicate apparent compensation of the decline in signal by thepredictive-kinetic method compared to the non-equilibrium, fixed-pointmethod. The data presented in FIG. 12 clearly show that during thenine-hour study, the magnitude of the predicted values is consistentlyhigher than that of the fixed-point method. The last four hours alsoshowed an increase in the ratios consistent with a more rapid decline inthe responses estimated by the fixed-point method. Hence, thepredictive-kinetic approach has the potential to measure essentially allglucose extracted into the hydrogel, regardless of the sensorsensitivity, thereby compensating for the signal decline seen with afixed integral measurement.

EXAMPLE 6 Compensation of Incomplete Reaction by Predicting Signal atCompletion

[0286] The data collection sequence used here was similar to that usedwith human subjects wearing the device. The sensors contained in thebiographer glucose monitor were preconditioned sequentially at 0.77 and0.42 V versus Ag/AgCl reference electrode (i.e., potential is appliedbetween the working electrode and a reference electrode (Ag/AgCl)) for10 minutes, respectively. Subsequent data collection was performed at0.42V. Multiple data points were collected from both sensors over a 7minute time period followed by a 3 minute off-period betweenmeasurements. Data points were obtained from each of two sensors bytaking a time point measurement on one sensor followed by a time pointmeasurement on the second sensor and repeating this switching back andforth over the 7 minute period. All measurements were performed at 32°C. temperature. To simulate glucose extraction, 3 μl of glucose of knownconcentration was deposited (two hours after the start ofpreconditioning) on each sensor attached to the biographer glucosemonitor in the middle of each 3 minute off-period. Sample addition wasrepeated two times, at four and six-hour intervals, resulting in sixreplicate measurements for a single glucose concentration. A similarprocedure was performed in a random fashion for the following glucoseconcentration: 0.00267, 0.00888, 0.015, 0.030, 0.0450, and 0.060 mM.Data were transferred to a computer for processing.

[0287] Typical time dependent responses of the biographer glucosemonitor to different concentrations of glucose are shown in FIG. 14. Themagnitude of the responses curves correlated to the concentration ofglucose samples. In standard practice, the values of the signals at 405seconds, which were inconsistent with values at complete reaction, wereused to perform a point-by-point subtraction of the time dependentresponse. Integration of the signal using this approach is shown in FIG.15. Though not apparent in FIG. 15, simply using measured current valueat 405 seconds for background subtraction resulted in over correction,as shown by non-equilibrium charge at latter time points.

[0288] Fits of curves, based on applying the predictive-kinetic methodof the present invention as represented in the embodiment of Eqn. 19, tothe responses in FIG. 15 are shown in FIG. 16. Fitting Eqn. 19 to thecharge versus time response (FIG. 16) compensated, to some extent, forthe over correction seen by simply using measured current value at 405seconds for background subtraction in FIG. 15.

[0289]FIG. 17 presents the data obtained from typical fits using Eqn. 21to fit the response data presented in FIG. 14.

S _(t) =S ₁ *e ^(k1*t) +S ₂ *e ^(k2*t)+final_Bkgrd  Eqn. 21

[0290] Even though data was collected for 405 seconds, the fitted lineswere extended to 1600 seconds to illustrate reliable estimate of a truecurrent at complete consumption of the glucose (i.e., end-point). FIG.18 presents integrated responses from fitted current after backgroundcorrection using the predicted, end-point, background current that wasobtained in FIG. 17. The profiles of charge values clearly demonstratethat a constant signal was achieved after 800 seconds for allconcentrations using this data processing method.

[0291] For all the concentrations investigated, the predicted backgroundcurrent determined by using Eqn. 21, was lower than or equal to themeasured value at 405 seconds. A plot of background current versusconcentration is presented in FIG. 19. In the figure, measured currentat 405 seconds showed a correlation with glucose concentration(y=380.47×+68.25; r²=0.5715), whereas predicted current was independentof glucose concentration (y=37.205x+67.53; r²=0.0106). These resultsdemonstrated incomplete consumption as a function of the amount ofglucose deposited on the sensors attached to the biographer glucosemonitor, which in turn affected any data processing option that used themeasured signal at 405 seconds for background correction. However, useof the predicted current at completion for background correctionresulted in higher sensitivity for the same glucose concentration andimproved the performance of the biographer glucose monitor.

[0292] A demonstration of higher sensitivity estimated by usingpredicted current from Eqn. 21 and integration of fitted response curveafter background correction with the resulting predicted backgroundvalue was shown by estimating the slope of a calibration curve betweenexperimental versus theoretical charge. The theoretical charge for eachconcentration of glucose was computed using the method presented inExample 2. The slope obtained using this method and imprecision for thismethod were compared to other data processing approaches and results areshown in Table 3. TABLE 3 Linearity Studies Comparison of Methods forFit of Models to Biographer Response Pooled Standard Fit Deviation RangeIntercept S_(y,x) (SD) Variance Model Method (sec) Slope (nC) R² (nC)(nC) Ratio Fidelity**** Fixed (not 0.4866 2088 0.9945  691  566 1.49 OKTime appli- at 405 cable) seconds* Eqn. 15-180 0.5349 2088 0.9957  676 538 1.58 OK 19** Eqn. 30-405 0.7485 2849 0.9936 1158 1078 1.15 OK 21***

[0293] Fixed time method represented use of the value at 405 seconds(FIG. 14) to perform background subtraction and integration of correctedcurrent. The charge value at 405 seconds represented the measured signal(FIG. 15). The results presented in Table 3 clearly demonstrated thatthe fit of current versus time curves, using Eqn. 21 to fit over therange of values from 30 to 405 seconds (FIG. 17), to predict signal atcompletion of the reaction, and integration of the fitted line afterbackground subtraction using the predicted value (FIG. 18) represented areliable and robust data processing method. The method employingend-point background values estimated using Eqn. 21 gave the largestslope (0.75) and showed that 75% of deposited glucose was accounted forby this method. Using Eqn.19 to fit integrated charge over 180 seconds(FIG. 16), after correction with background signal measured at 405seconds, accounted for 53% of the glucose sample. The fixed time methodonly accounted for about 49% of the glucose sample.

[0294] The variance ratios for each method were estimated and theresults are shown in Table 3. Analysis of these values using an F-testshowed that the three models are valid for estimating responses of thebiographer glucose monitor to glucose (i.e., computed F-ratio for thethree models are less than the F-value at 95% confidence interval, i.e.,less than 2.76). More importantly, fits of Eqn. 21 to the current versustime response and subsequent data treatment as described herein allowedfor a reliable estimate of equilibrium value consistent with completeconsumption of the glucose. Because this method estimated total glucoseconsumed, it provides an invaluable tool to examine decline insensitivity of the response of the biographer glucose monitor to glucoseover an extended period.

[0295] As is apparent to one of skill in the art, various modificationand variations of the above embodiments can be made without departingfrom the spirit and scope of this invention. Such modifications andvariations are within the scope of this invention.

What is claimed is:
 1. A method for measuring glucose present in asubject, said method comprising: (A) transdermally extracting a samplecomprising glucose from the subject using a sampling system that is inoperative contact with a skin or mucosal surface of said subject; (B)obtaining a measured signal over time, comprising a measured signalresponse curve, from the extracted glucose, wherein said measured signalis specifically related to the amount or concentration of glucose, andsaid measured signal response curve comprises kinetic and equilibriumregions; (C) using (i) a mathematical model comprising selectedparameters, wherein said model describes the measured signal responsecurve, and said mathematical model is selected from the group consistingof a first order process, combined first order and zero order process, aparallel multiple first order process, a flux process, and an n^(th)order process, and (ii) an error minimization method, to iterativelyestimate values of the parameters using said model and errorminimization method to fit a predicted response curve to said measuredsignal response curve, wherein (a) the error minimization methodprovides a calculated error based on differences between said predictedand measured signal response curves, and (b) said estimating isiteratively performed until the calculated error between the predictedand measured signal response curves falls within an acceptable range oruntil no further statistically significant change is seen in thecalculated error, at which time iterative estimation of the parametersis stopped, said iterative estimation and error minimization results ina predicted response curve corresponding to said measured signalresponse curve, said predicted response curve yields a predictedend-point value and a measurement correlated to the amount orconcentration of the glucose.
 2. The method of claim 1, wherein saidmeasured signal response curve comprises a measurement of current overtime, or measurement of charge over time.
 3. The method of claim 2,wherein said measured signal response curve comprises a measurement ofcurrent over time, said predicted end-point value is an estimated signalat equilibrium, where said predicted end-point value provides apredicted final background value, and said measurement correlated to theamount or concentration of glucose corresponds to an area under thepredicted response curve.
 4. The method of claim 3, wherein said areaunder the predicted response curve is obtained by integration of thepredicted response curve.
 5. The method of claim 4, wherein before saidintegration is performed said final background value is used to performa background subtraction correction of the predicted response curve andsaid measurement correlated to the amount or concentration of glucosecorresponds to an area under the predicted response curve.
 6. The methodof claim 4, wherein the end-point value of the integrated predictedresponse curve is converted to an amount or concentration of theglucose.
 7. The method of claim 5, wherein the end-point value of theintegrated predicted response curve is converted to an amount orconcentration of the glucose.
 8. The method of claim 1, wherein themathematical model further comprises a zero-order component.
 9. Themethod of claim 6, wherein conversion of the end-point value of theintegrated predicted response curve to an amount or concentration ofglucose is performed by a method comprising applying a calibrationvalue.
 10. The method of claim 1, wherein said mathematical modelcomprises more than one process and each process comprises selectedparameters.
 11. The method of claim 10, wherein each process has acorresponding weighting factor.
 12. The method of claim 1, wherein abackground subtraction is performed on the measured signal responsecurve before (C) is performed.
 13. The method of claim 1, wherein (A),(B), and (C) are performed at least two times to obtain a series ofmeasurements.
 14. The method of claim 13, wherein after estimation ofeach predicted response curve for each measured signal response curve inthe series of measurements an amount or concentration of the glucose isdetermined based on the predicted response curve.
 15. The method ofclaim 1, wherein said measured signal response curve comprises datapoints.
 16. The method of claim 15, wherein at least three data pointsare obtained from the kinetic region of the measured signal responsecurve, and these data points are used to estimate the half-life of themeasured signal.
 17. The method of claim 16, wherein said obtaining ofthe measured signal continues for a time period corresponding to atleast three half-lives of the signal.
 18. The method of claim 1, whereinsaid obtaining is carried out for a defined period of time.
 19. Themethod of claim 1, wherein (B) further comprises integration of themeasured signal response curve before using the mathematical model tofit the predicted signal response curve to the measured signal responsecurve.
 20. The method of claim 1, wherein (C) further comprisesintegration of the predicted response curve after using the mathematicalmodel to fit the predicted signal response curve to the measured signalresponse curve.
 21. The method of claim 1, wherein said mathematicalmodel comprises a first order process.
 22. The method of claim 21,wherein said first order process comprises the following: S _(t) =S_(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1) where S_(o), S_(t), and S_(∞)are initial, intermediate, and end-point signals, k and t are theobserved first-order rate constant and time, respectively.
 23. Themethod of claim 1, wherein said mathematical model comprises a parallelmultiple first order process.
 24. The method of claim 23, wherein saidparallel multiple first order process comprises the following: S _(t) =S_(o)+(S _(∞1) −S _(o))*(1−e ^(−k1t))+(S _(∞2) −S _(o))*(1−e ^(−k2t))+(S_(∞3) −S _(o))*(1−e ^(−k3t))+ . . .  (Eqn. 6A) where S_(o), S_(t) areinitial and intermediate signals, S_(∞1), S_(∞2), S_(∞3), etc., areend-point signals (related to k₁, k₂, k₃, etc., respectively), k₁, k₂,k₃, etc., are the observed first-order rate constants, and t is time.25. The method of claim 24, wherein the predicted end-point value isdescribed by the following equation S _(∞)=(S _(∞1) +S _(∞2) +S _(∞3)+ .. . )+S _(o)  (Eqn. 6B)
 26. The method of claim 24, wherein a change inthe predicted end-point value relative to the initial signal isdescribed by the following equation ΔS _(∞)=(S _(∞1) +S _(∞2) +S _(∞3)+. . . )  (Eqn. 6C)
 27. The method of claim 23, wherein said parallelmultiple first order process comprises the following: S _(t) =S _(o)+(S_(∞1) −S _(o))*(1−e ^(−k1t))+(S _(∞2) −S _(o))*(1−e ^(−k2t))  (Eqn. 10)where S_(o), and S_(t), are initial and intermediate signals, S_(∞1),and S_(∞2) are end-point signals (related to k₁ and k₂, respectively),k₁, k₂, and t are the observed first-order rate constants and time. 28.The method of claim 24, wherein said parallel multiple first orderprocess further comprises at least one zero order process.
 29. Themethod of claim 28, wherein said parallel multiple first order processcomprises the following: S _(t) =S _(o) +k _(o) t+(S _(∞1) −S _(o))*(1−e^(−k1t))+(S _(∞2) −S _(o))*(1−e ^(−k2t))+(S _(∞3) −S _(o))*(1−e^(−k3t))  (Eqn. 16) where S_(o), S_(t) are initial and intermediatesignals, S_(∞1), S_(∞2), S_(∞3), are end-point signals (related to k₁,k₂, k₃, respectively), k₁, k₂, k₃, are the observed first-order rateconstants, k_(o) is a zero order rate constant, and t is time.
 30. Themethod of claim 23, wherein said parallel multiple first order processfurther comprises at least one quadratic or square root term.
 31. Themethod of claim 3, wherein said mathematical model comprises a parallelmultiple first order process.
 32. The method of claim 31, wherein saidparallel multiple first order process comprises the following: S _(t) =S₀ +S ₁ *e ^(−k1*t) +S ₂ *e ^(−k2*t)+final_Bkgrd  Eqn. 20 where S₀ isresponse at t=0, t is time, S_(t) is a total signal at time t, S₁ and S₂are signals at time t consistent with two processes associated withapparent rate constants k₁ and k₂, and final_bkgrd is an estimatedsignal response at completion of a reaction used to obtain the measuredsignal.
 33. The method of claim 32, wherein the area under the predictedresponse curve is obtained by integration.
 34. The method of claim 33,wherein before said integration is performed said final_bkgrd value isused to perform a background subtraction correction of the predictedresponse curve and said measurement correlated to the amount orconcentration of glucose corresponds to the area under the predictedresponse curve.
 35. The method of claim 31, wherein said parallelmultiple first order process comprises the following: S _(t) =S ₁ *e^(−k1*t) +S ₂ *e ^(−k2*t)+final_Bkgrd  Eqn. 21 where t is time, S_(t) isa total signal at time t, S₁ and S₂ are signals at time t consistentwith two processes associated with apparent rate constants k₁ and k₂,and final_bkgrd is an estimated signal response at completion of areaction used to obtain the measured signal.
 36. The method of claim 35,wherein the area under the predicted response curve is obtained byintegration.
 37. The method of claim 36, wherein before said integrationis performed said final_bkgrd value is used to perform a backgroundsubtraction correction of the predicted response curve and saidmeasurement correlated to the amount or concentration of glucosecorresponds to the area under the predicted response curve.
 38. Themethod of claim 1, wherein said mathematical model comprises an n^(th)order process.
 39. The method of claim 38, wherein said n^(th) orderprocess comprises the following: S _(t) =S _(∞)(−/+)[k(n−1)*t(+/−)(S_(∞) −S _(o))^(1−n)]^(1/(1−n))  (Eqn. 8) where S_(o), S_(t), and S_(∞)are initial, intermediate, and end-point signals, k and t are theobserved rate constant and time, n is the order of the process, where ndoes not equal 1, and for (−/+) the first function (−) is used for datathat increase in magnitude as a function of time, and the secondfunction (+) is used for the reverse case, correspondingly for (+/−) thefirst function (+) is used for data that increase in magnitude as afunction of time, and the second function (−) is used for the reversecase.
 40. The method of claim 13, wherein for different measurements inthe series different mathematical models are selected to estimate apredicted end-point value.
 41. The method of claim 13, wherein a singlemathematical model is selected to estimate the predicted end-pointvalues for all measurements in the series.
 42. The method of claim 16,wherein the estimate of the half-life (t_(½)) further comprises,estimating a rate constant (k) for a first order model comprising S _(t)=S _(∞)−(S _(∞) −S _(o))e ^(−kt)  (Eqn. 1) where S_(o), S_(t), and S_(∞)are initial, intermediate, and end-point signals, k and t are theobserved first-order rate constant and time, respectively, whereinestimating said rate constant is performed by a method comprisingplotting the natural log of signal (S_(t)−S_(o)) over time, where theslope of the resulting line corresponds to an estimate of k, and thehalf-life of the signal is calculated by using t_(½)=ln 2/k.
 43. Themethod of claim 1, wherein the glucose is extracted from the subject asin (A) into a collection reservoir to obtain a concentration of theglucose in said reservoir.
 44. The method of claim 43, wherein thecollection reservoir is in contact with the skin or mucosal surface ofthe subject and the glucose is extracted using an iontophoretic currentapplied to said skin or mucosal surface.
 45. The method of claim 44,wherein the collection reservoir comprises an enzyme that reacts withthe extracted glucose to produce an electrochemically detectable signal.46. The method of claim 45, wherein the enzyme comprises glucoseoxidase.
 47. The method of claim 1, wherein said mathematical modelcomprises a flux model.
 48. The method of claim 47, wherein said fluxmodel comprises the following: $\begin{matrix}{S_{t} = {S_{o} + {\left( {S_{\infty} - S_{o}} \right)\left\lbrack {1 + {2{\sum\limits_{i = 0}^{\infty}{\left( {- 1} \right)^{i}{\exp \left( {{- k_{i}}t} \right)}}}}} \right\rbrack}}} & \left( {{Eqn}.\quad 9} \right)\end{matrix}$

where S_(o), S_(t), and S_(∞) are initial, intermediate, and final (orend-point) signals, k_(i)=ki²π², k is the characteristic diffusion rateconstant, t is time, and i is a dummy-variable.
 49. The method of claim1, wherein said transdermal extraction is sonophoretic.
 50. A method formeasuring glucose present in a subject, said method comprising: (a)transdermally extracting a sample comprising the glucose from thesubject using a sampling system that is in operative contact with a skinor mucosal surface of said subject; (b) obtaining a measured signal overtime, comprising a measured signal response curve, from the extractedglucose, wherein said measured signal is specifically related to theamount or concentration of glucose, and said measured signal responsecurve comprises kinetic and equilibrium regions; (c) selecting amathematical model comprising selected parameters, wherein said modeldescribes the measured signal response curve, and said mathematicalmodel is selected from the group consisting of a first order process,combined first order and zero order process, a parallel multiple firstorder process, a flux process, and an n^(th) order process; (d)iteratively estimating values of the parameters using said model and anerror minimization method to fit a predicted response curve to saidmeasured signal response curve, wherein (i) the error minimizationmethod provides a calculated error based on differences between saidpredicted and measured signal response curves, and (ii) said estimatingis iteratively performed until the calculated error between thepredicted and measured signal response curves falls within an acceptablerange or until no further statistically significant change is seen inthe calculated error, at which time iterative estimation of theparameters is stopped, said iterative estimation and error minimizationresults in a predicted response curve corresponding to said measuredsignal response curve, said predicted response curve yields a predictedend-point value and a measurement correlated to the amount orconcentration of the glucose.
 51. The method of claim 50, wherein (a),(b), and (d) are performed at least two times to obtain a series ofmeasurements.
 52. The method of claim 51, wherein after each (d) themeasurement correlated to the amount or concentration of the glucose isconverted to an amount or concentration of glucose.
 53. The method ofclaim 51, wherein for different measurements in the series differentmathematical models are selected to estimate a predicted end-pointvalue.
 54. The method of claim 51, wherein a single mathematical modelis selected to estimate the predicted end-point values for allmeasurements in the series.
 55. A method for measuring glucose presentin a subject, said method comprising: (A) transdermally extracting asample comprising glucose from the subject using a sampling system thatis in operative contact with a skin or mucosal surface of said subject;(B) obtaining a measured current signal over time, comprising a measuredcurrent signal response curve, from the extracted glucose, wherein saidmeasured current signal is specifically related to the amount orconcentration of glucose, and said measured current signal responsecurve comprises kinetic and equilibrium regions; (C) using (i) amathematical model comprising selected parameters, wherein said modeldescribes the measured current signal response curve, and saidmathematical model is selected from the group consisting of a firstorder process, combined first order and zero order process, a parallelmultiple first order process, a flux process, and an n^(th) orderprocess, and (ii) an error minimization method, to iteratively estimatevalues of the parameters using said model and error minimization methodto fit a predicted response curve to said measured current signalresponse curve, wherein (a) the error minimization method provides acalculated error based on differences between said predicted andmeasured signal response curves, and (b) said estimating is iterativelyperformed until the calculated error between the predicted and measuredsignal response curves falls within an acceptable range or until nofurther statistically significant change is seen in the calculatederror, at which time iterative estimation of the parameters is stopped,said iterative estimation and error minimization results in a predictedresponse curve corresponding to said measured signal response curve,said predicted response curve yields a predicted end-point value and ameasurement correlated to the amount or concentration of the glucose;(D) performing a background subtraction correction of the predictedresponse curve using the predicted end-point value as a final backgroundvalue; and (E) integrating the background corrected predicted responsecurve to obtain a measurement of the amount or concentration of glucosein the subject at the time of sampling.
 56. A method for compensatingfor an incomplete reaction involving the detection of an analyte bypredicting a background signal, said method comprising (A) providing ameasured signal over time, comprising a measured signal response curve,wherein said measured signal is specifically related to an amount orconcentration of analyte, and said measured signal response curvecomprises kinetic and equilibrium regions; (B) using (i) a mathematicalmodel comprising selected parameters, wherein said model describes themeasured signal response curve, and said mathematical model is selectedfrom the group consisting of a first order process, combined first orderand zero order process, a parallel multiple first order process, a fluxprocess, and an n^(th) order process, and (ii) an error minimizationmethod, to iteratively estimate values of the parameters using saidmodel and error minimization method to fit a predicted response curve tosaid measured signal response curve, wherein (a) the error minimizationmethod provides a calculated error based on differences between saidpredicted and measured signal response curves, and (b) said estimatingis iteratively performed until the calculated error between thepredicted and measured signal response curves falls within an acceptablerange or until no further statistically significant change is seen inthe calculated error, at which time iterative estimation of theparameters is stopped, said iterative estimation and error minimizationresults in a predicted response curve corresponding to said measuredsignal response curve, said predicted response curve yields a predictedfinal background value and a measurement correlated to the amount orconcentration of the analyte; and (C) performing a backgroundsubtraction correction of the predicted response curve using thepredicted final background value, wherein said background subtractioncompensates for an incomplete reaction involved in the detection of theanalyte.
 57. The method of claim 56, wherein said measured signalresponse curve comprises a measurement of current over time, saidpredicted final background value is an estimate of signal at completionof the reaction, and said measurement correlated to the amount orconcentration of analyte corresponds to an area under the correctedpredicted response curve.
 58. The method of claim 57, wherein said areaunder the corrected predicted response curve is obtained by integrationof the corrected predicted response curve.
 59. The method of claim 58,wherein an end-point value of the integrated predicted response curve isconverted to an amount or concentration of the analyte.
 60. One or moremicroprocessors, comprising programming to control (i) a measurementcycle comprising (a) operating a sampling device for extracting a samplefrom the biological system, said sample comprising glucose, and (b)operating a sensing device for obtaining a measured signal over time,comprising a measured signal response curve, from the extracted glucose,wherein said measured signal is specifically related to the amount orconcentration of glucose, and said measured signal response curvecomprises kinetic and equilibrium regions; and (ii) a computation methodusing (a) a mathematical model comprising selected parameters, whereinsaid model describes the measured signal response curve, and saidmathematical model is selected from the group consisting of a firstorder process, combined first order and zero order process, a parallelmultiple first order process, a flux process, and an n^(th) orderprocess, and (b) an error minimization method, to iteratively estimatevalues of the parameters using said model and error minimization methodto fit a predicted response curve to said measured signal responsecurve, wherein (a′) the error minimization method provides a calculatederror based on differences between said predicted and measured signalresponse curves, and (b′) said estimating is iteratively performed untilthe calculated error between the predicted and measured signal responsecurves falls within an acceptable range or until no furtherstatistically significant change is seen in the calculated error, atwhich time iterative estimation of the parameters is stopped, saiditerative estimation and error minimization results in a predictedresponse curve corresponding to said measured signal response curve,said predicted response curve yields a predicted end-point value and ameasurement correlated to the amount or concentration of the glucose.61. The one or more microprocessors of claim 60, further programmed (i)to perform a series of measurement cycles resulting in a series ofmeasured signal response curves, and (ii) to provide a predictedresponse curve corresponding to each measured signal response curve. 62.The one or more microprocessors of claim 60, wherein said measuredsignal response curve comprises a measurement of current over time, ormeasurement of charge over time.
 63. The one or more microprocessors ofclaim 60, wherein said measured signal response curve comprises ameasurement of current over time, said predicted end-point value is anestimated signal at equilibrium, where said predicted end-point valueprovides a predicted final background value, and said measurementcorrelated to the amount or concentration of glucose corresponds to anarea under the predicted response curve
 64. The one or moremicroprocessors of claim 63, wherein said area under the predictedresponse curve is obtained by integration of the predicted responsecurve.
 65. The one or more microprocessors of claim 64, wherein beforesaid integration is performed said final background value is used toperform a background subtraction correction of the predicted responsecurve and said measurement correlated to the amount or concentration ofglucose corresponds to an area under the predicted response curve. 66.The one or more microprocessors of claim 64, wherein the end-point valueof the integrated predicted response curve is converted to an amount orconcentration of the glucose.
 67. The one or more microprocessors ofclaim 65, wherein the end-point value of the integrated predictedresponse curve is converted to an amount or concentration of theglucose.
 68. The one or more microprocessors of claim 60, wherein thesampling device comprises one or more collection reservoirs into whichthe sample is collected.
 69. The one or more microprocessors of claim68, wherein the sampling device comprises an iontophoretic device toextract the sample comprising glucose from the subject into at least onecollection reservoir.
 70. The one or more microprocessors of claim 68,wherein the collection reservoir comprises an enzyme that reacts withthe extracted glucose to produce an electrochemically detectable signal.71. The one or more microprocessors of claim 70, wherein the enzymecomprises glucose oxidase.
 72. The one or more microprocessors of claim60, wherein the sampling device comprises a laser device.
 73. Amonitoring system comprising the one or more microprocessors of claim60, wherein said monitoring system further comprises a sampling deviceand a sensing device.
 74. A monitoring system for frequent measurementof glucose amount or concentration present in a subject, said systemcomprising, in operative combination: (A) a sampling device forfrequently extracting a sample comprising glucose from the subject,wherein said sampling device is adapted for extracting the glucoseacross a skin or mucosal surface of said subject; (B) a sensing devicein operative contact with the glucose extracted by the sampling device,wherein said sensing device obtains a measured signal over time,comprising a measured signal response curve, from the extracted glucose,wherein said measured signal is specifically related to the amount orconcentration of glucose, and said measured signal response curvecomprises kinetic and equilibrium regions; (C) one or moremicroprocessor(s) in operative communication with the sampling deviceand the sensing device, wherein said microprocessor is capable of (i)controlling the sampling device and the sensing device to obtain aseries of measured signals, in the form of measured signal responsecurves, at selected time intervals, (ii) predicting measurement valuesfor each measured signal in the series by employing (a) a mathematicalmodel comprising selected parameters, wherein said model describes themeasured signal response curve of (B), and said mathematical model isselected from the group consisting of a first order process, combinedfirst order and zero order process, a parallel multiple first orderprocess, a flux process, and an n^(th) order process, and (b) an errorminimization method, to iteratively estimate values of the parametersusing said model and error minimization method to fit a predictedresponse curve to said measured signal response curve, wherein (a′) theerror minimization method provides a calculated error based ondifferences between said predicted and measured signal response curves,and (b′) said estimating is iteratively performed until the calculatederror between the predicted and measured signal response curves fallswithin an acceptable range or until no further statistically significantchange is seen in the calculated error, at which time iterativeestimation of the parameters is stopped, said iterative estimation anderror minimization results in a predicted response curve correspondingto said measured signal response curve, said predicted response curveyields a predicted end-point value and a measurement correlated to theamount or concentration of the glucose, and (iii) converting eachmeasurement correlated to the amount or concentration of the glucose inthe series to a measurement value indicative of the amount orconcentration of glucose present in the subject.
 75. The monitoringsystem of claim 74, wherein said measured signal response curvecomprises a measurement of current over time, or measurement of chargeover time.
 76. The monitoring system of claim 75, wherein said measuredsignal response curve comprises a measurement of current over time, saidpredicted end-point value is an estimated signal at equilibrium, wheresaid predicted end-point value provides a predicted final backgroundvalue, and said measurement correlated to the amount or concentration ofglucose corresponds to an area under the predicted response curve. 77.The monitoring system of claim 76, wherein said area under the predictedresponse curve is obtained by integration of the predicted responsecurve.
 78. The monitoring system of claim 77, wherein before saidintegration is performed said final background value is used to performa background subtraction correction of the predicted response curve andsaid measurement correlated to the amount or concentration of glucosecorresponds to an area under the predicted response curve.
 79. Themonitoring system of claim 77, wherein the end-point value of theintegrated predicted response curve is converted to an amount orconcentration of the glucose.
 80. The monitoring system of claim 78,wherein the end-point value of the integrated predicted response curveis converted to an amount or concentration of the glucose.
 81. Themonitoring system of claim 74, wherein the sampling device comprises oneor more collection reservoirs into which the sample is collected. 82.The monitoring system of claim 81, wherein the sampling device comprisesan iontophoretic device to extract the sample comprising glucose fromthe subject into at least one collection reservoir.
 83. The monitoringsystem of claim 82, wherein the collection reservoir comprises an enzymethat reacts with the extracted glucose to produce an electrochemicallydetectable signal.
 84. The monitoring system of claim 83, wherein theenzyme comprises glucose oxidase.
 85. The monitoring system of claim 74,wherein the sampling device comprises a laser device.